SIMPLICIAL MATRIX-TREE THEOREMS

arXiv:0802.2576v2 [math.CO] 21 Aug 2008

SIMPLICIAL MATRIX-TREE THEOREMS ART M. DUVAL, CAROLINE J. KLIVANS, AND JEREMY L. MARTIN Abstract. We generalize the definition and enumeration of spanning trees from the setting of graphs to that of arbitrary-dimensional simplicial complexes ∆, extending an idea due to G. Kalai. We prove a simplicial version of the Matrix-Tree Theorem that counts simplicial spanning trees, weighted by the squares of the orders of their top-dimensional integral homology groups, in terms of the Laplacian matrix of ∆. As in the graphic case, one can obtain a more finely weighted generating function for simplicial spanning trees by assigning an indeterminate to each vertex of ∆ and replacing the entries of the Laplacian with Laurent monomials. When ∆ is a shifted complex, we give a combinatorial interpretation of the eigenvalues of its weighted Laplacian and prove that they determine its set of faces uniquely, generalizing known results about threshold graphs and unweighted Laplacian eigenvalues of shifted complexes.

1. Introduction This article is about generalizing the Matrix-Tree Theorem from graphs to simplicial complexes. 1.1. The classical Matrix-Tree Theorem. We begin by reviewing the classical case; for a more detailed treatment, see, e.g., [8]. Let G be a finite, simple, undirected graph with vertices V (G) = [n] = {1, 2, . . . , n} and edges E(G). A spanning subgraph of G is a graph T with V (T ) = V (G) and E(T ) ⊆ E(G); thus a spanning subgraph may be specified by its edge set. A spanning subgraph T is a spanning tree if (a) T is acyclic; (b) T is connected; and (c) |E(T )| = |V (T )|−1. It is a fundamental property of spanning trees (the “two-out-of-three theorem”) that any two of these three conditions together imply the third. The Laplacian of G is the n × n symmetric matrix L = L(G) with entries   degG (i) if i = j, Lij = −1 if i, j are adjacent,   0 otherwise,

where degG (i) is the degree of vertex i (the number of edges having i as an endpoint). Equivalently, L = ∂∂ ∗ , where ∂ is the (signed) vertex-edge incidence matrix and ∂ ∗ is its transpose. If we regard G as a one-dimensional simplicial complex, then ∂ is just the simplicial boundary map from 1-faces to 0-faces, and ∂ ∗ is the simplicial coboundary map. The matrix L is symmetric, hence diagonalizable, so it has n real eigenvalues (counting multiplicities). The number of nonzero eigenvalues of L is n − c, where c is the number of components of G. The Matrix-Tree Theorem, first observed by Kirchhoff [22] in his work on electrical circuits (modern references include [8], [29] and [34, Chapter 5]), expresses the number τ (G) of spanning trees of G in terms of L. The theorem has two equivalent formulations. Theorem 1.1 (Classical Matrix-Tree Theorem). Let G be a connected graph with n vertices, and let L be its Laplacian matrix. Date: August 14, 2008. 2000 Mathematics Subject Classification. Primary 05A15; Secondary 05E99, 05C05, 05C50, 15A18, 57M15. Key words and phrases. Simplicial complex, spanning tree, tree enumeration, Laplacian, spectra, eigenvalues, shifted complex. Second author partially supported by NSF VIGRE grant DMS-0502215. Third author partially supported by an NSA Young Investigators Grant. 1

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ART M. DUVAL, CAROLINE J. KLIVANS, AND JEREMY L. MARTIN

(1) If the eigenvalues of L are λ0 = 0, λ1 , . . . , λn−1 , then τ (G) =

λ1 · · · λn−1 . n

(2) For 1 ≤ i ≤ n, let Li be the reduced Laplacian obtained from L by deleting the ith row and ith column. Then τ (G) = det Li . Well-known corollaries of the Matrix-Tree Theorem include Cayley’s formula [9] τ (Kn ) = nn−2

(1)

where Kn is the complete graph on n vertices, and Fiedler and Sedl´ açek’s formula [16] τ (Kn,m ) = nm−1 mn−1

(2)

where Kn,m is the complete bipartite graph on vertex sets of sizes n and m. The Matrix-Tree Theorem can be refined by introducing an indeterminate eij = eji for each pair ˆ is then of vertices i, j, setting eij = 0 if i, j do not share a common edge. The weighted Laplacian L defined as the n × n matrix with entries P n   k=1 eik if i = j, ˆ Lij = −eij if i, j are adjacent,   0 otherwise. ˆ Theorem 1.2 (Weighted Matrix-Tree Theorem). Let G be a graph with n vertices, and let L be its weighted Laplacian matrix. ˆ0 = 0, λ ˆ1, . . . , λ ˆn−1 , then ˆ are λ (1) If the eigenvalues of L X

Y

eij =

T ∈T (G) ij∈T

ˆ1 · · · λ ˆn−1 λ n

where T (G) is the set of all spanning trees of G. ˆ i be the reduced weighted Laplacian obtained from L ˆ by deleting the ith (2) For 1 ≤ i ≤ n, let L th row and i column. Then X Y ˆ i. eij = det L T ∈T (G) ij∈T

By making appropriate substitutions for the indeterminates eij , it is often possible to obtain finer enumerative information than merely the number of spanning trees. For instance, when G = Kn , introducing indeterminates x1 , . . . , xn and setting eij = xi xj for all i, j yields the Cayley-Pr¨ ufer Theorem, which enumerates spanning trees of Kn by their degree sequences: X

n Y

degT (i)

xi

= x1 · · · xn (x1 + · · · + xn )n−2 .

(3)

T ∈T (G) i=1

Note that Cayley’s formula (1) can be recovered from the Cayley-Pr¨ ufer Theorem by setting x1 = · · · = xn = 1. 1.2. Simplicial spanning trees and how to count them. To extend the scope of the MatrixTree Theorem from graphs to simplicial complexes, we must first say what “spanning tree” means in arbitrary dimension. Kalai [20] proposed a definition that replaces the acyclicity, connectedness,

SIMPLICIAL MATRIX-TREE THEOREMS

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and edge-count conditions with their analogues in simplicial homology. Our definition adapts Kalai’s definition to a more general class of simplicial complexes.1 Let ∆ be a d-dimensional simplicial complex, and let Υ ⊂ ∆ be a subcomplex containing all faces of ∆ of dimension < d. We say that Υ is a simplicial spanning tree of ∆ if the following three conditions hold: ˜ d (Υ, Z) = 0, H (4a) ˜ d−1 (Υ, Z)| < ∞, and |H fd (Υ) = fd (∆) − β˜d (∆) + β˜d−1 (∆),

(4b) (4c)

˜ i denotes reduced simplicial homology (for which see, e.g., [18, §2.1]). (The conditions (4a) where H and (4b) were introduced by Kalai in [20], while (4c) is more general, as we will explain shortly.) When d = 1, the conditions (4a). . . (4c) say respectively that Υ is acyclic, connected, and has one fewer edge than it has vertices, recovering the definition of the spanning tree of a graph. Moreover, as we will show in Proposition 3.5, any two of the three conditions together imply the third. A graph G has a spanning tree if and only if G is connected. The corresponding condition for a ˜ i (∆, Q) = 0 for all i < d; that is, ∆ has the rational simplicial complex ∆ of dimension d is that H homology type of a wedge of d-dimensional spheres. We will call such a complex acyclic in positive codimension, or APC for short. This condition, which we will assume throughout the rest of the introduction, is much weaker than Cohen-Macaulayness (by Reisner’s theorem [30]), and therefore encompasses many complexes of combinatorial interest, including all connected graphs, simplicial spheres, shifted, matroid, and Ferrers complexes, and some chessboard and matching complexes. For k ≤ d, let ∂ = ∂k be the kth simplicial boundary matrix of ∆ (with rows and columns indexed respectively by (k − 1)-dimensional and k-dimensional faces of ∆), and let ∂ ∗ be its transpose. The (kth up-down) Laplacian of ∆ is L = ∂∂ ∗ ; this can be regarded either as a square matrix of size fk−1 (∆) or as a linear endomorphism on (k − 1)-chains of ∆. Define invariants πk = πk (∆) = product of all nonzero eigenvalues of L, X ˜ k−1 (Υ)|2 , |H τk = τk (∆) = Υ∈Tk (∆)

where Tk (∆) denotes the set of all k-trees of ∆ (that is, simplicial spanning trees of the k-skeleton of ∆). Kalai [20] studied these invariants in the case that ∆ is a simplex on n vertices, and proved the formula n−2 (5) τk (∆) = n( k ) (of which Cayley’s formula (1) is the special case k = 1). Kalai also proved a natural weighted analogue of (5) enumerating simplicial spanning trees by their degree sequences, thus generalizing the Cayley-Pr¨ ufer Theorem (3). Given disjoint vertex sets V1 , . . . , Vr (“color classes”), the faces of the corresponding complete colorful complex Γ are those sets of vertices with no more than one vertex of each color. Equivalently, Γ is the simplicial join V1 ∗ V2 ∗ · · · ∗ Vr of the 0-dimensional complexes Vi . Adin [1] extended Kalai’s work by proving a combinatorial formula for τk (Γ), which we shall not reproduce here, for every 1 ≤ k < r. Note that when r = 2, the complex Γ is a complete bipartite graph, and if |Vi | = 1 for all i, then Γ is a simplex. Thus both (2) and (5) can be recovered from Adin’s formula. Kalai’s and Adin’s beautiful formulas inspired us to look for more results about simplicial spanning tree enumeration, and in particular to formulate a simplicial version of the Matrix-Tree Theorem 1There are many other definitions of “simplicial tree” in the literature, depending on which properties of trees one

wishes to extend; see, e.g., [4, 10, 15, 19, 28]. By adopting Kalai’s idea, we choose a definition that lends itself well to enumeration. The closest to ours in spirit is perhaps that of Masbaum and Vaintrob [28], whose main result is a Matrix-Tree-like theorem enumerating a different kind of 2-dimensional tree using Pfaffians rather than Laplacians.

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that could be applied to as broad a class of complexes as possible. Our first main result generalizes the Matrix-Tree Theorem to all APC simplicial complexes. Theorem 1.3 (Simplicial Matrix-Tree Theorem). Let ∆ be a d-dimensional APC simplicial complex. Then: (1) We have πd (∆) =

τd (∆)τd−1 (∆) . ˜ d−2 (∆)|2 |H

(2) Let U be the set of facets of a (d− 1)-SST of ∆, and let LU be the reduced Laplacian obtained by deleting the rows and columns of L corresponding to U . Then τd (∆) =

˜ d−2 (∆)|2 |H det LU . ˜ d−2 (∆U )|2 |H

We will prove these formulas in Section 4. In the special case d = 1, the number τ1 (∆) is just the number of spanning trees of the graph ∆, recovering the classical Matrix-Tree Theorem. When d ≥ 2, there can exist spanning trees with finite but nontrivial homology groups (the simplest example is the real projective plane). In this case, τk (∆) is greater than the number of spanning trees, because these “torsion trees” contribute more than 1 to the count. This phenomenon was first observed by Bolker [7], and arises also in the study of cyclotomic matroids [26] and cyclotomic polytopes [3]. The Weighted Matrix-Tree Theorem also has a simplicial analogue. Introduce an indeterminate xF for each facet (maximal face) F ∈ ∆, and for every set T of facets define monomials xT = Q 2 ˆ F ∈T xF and XT = xT . Construct the weighted boundary matrix ∂ by multiplying each column of ∂ by xF , where F is the facet of ∆ corresponding to that column. Let π ˆk be the product of the ˆ ud nonzero eigenvalues of L , and let ∆,k−1 X ˜ k−1 (Υ)|2 XΥ . |H τˆk = τˆk (∆) = Υ∈Tk (∆)

Theorem 1.4 (Weighted Simplicial Matrix-Tree Theorem). Let ∆ be a d-dimensional APC simplicial complex. Then: (1) We have2 π ˆd (∆) =

τˆd (∆)τd−1 (∆) . ˜ d−2 (∆)|2 |H

ˆ U be the reduced Laplacian obtained (2) Let U be the set of facets of a (d− 1)-SST of ∆, and let L ˆ by deleting the rows and columns of L corresponding to U . Then τˆd (∆) =

˜ d−2 (∆)|2 |H Û . det L ˜ d−2 (∆U )|2 |H

We will prove these formulas in Section 5. Setting xF = 1 for all F in Theorem 1.4 recovers Theorem 1.3. In fact, more is true; setting xF = 1 in the multiset of eigenvalues of the weighted Laplacian (reduced or unreduced) yields the eigenvalues of the corresponding unweighted Laplacian. If the complex ∆ is Laplacian integral, that is, its Laplacian matrix has integer eigenvalues, then we can hope to find a combinatorial interpretation of the factorization of τˆd (∆) furnished by Theorem 1.4. An important class of Laplacian integral simplicial complexes are the shifted complexes. 2Despite appearances, there are no missing hats on the right-hand side of this formula! Only τ (∆) has been d

replaced with its weighted analogue; τd−1 (∆) is still just an integer.


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1.3. Results on shifted complexes. Let p ≤ q be integers, and let [p, q] = {i ∈ Z : p ≤ i ≤ q}. A simplicial complex Σ on vertex set [p, q] is shifted if the following condition holds: whenever i < j are vertices and F ∈ Σ is a face such that i 6∈ F and j ∈ F , then F \{j} ∪ {i} ∈ Σ. Equivalently, define the componentwise partial order on finite sets of positive integers as follows: A B whenever A = {a1 < · · · < am }, B = {b1 < · · · < bm }, and ai ≤ bi for all i. Then a complex is shifted precisely when it is an order ideal with respect to the componentwise partial order. (See [33, chapter 3] for general background on partially ordered sets.) Shifted complexes were used by Björner and Kalai [5] to characterize the f -vectors and Betti numbers of all simplicial complexes. Shifted complexes are also one of a small handful of classes of simplicial complexes whose Laplacian eigenvalues are known to be integral. In particular, Duval and Reiner [13, Thm. 1.1] proved that the Laplacian eigenvalues of a shifted complex Σ on [p, q] are given by the conjugate of the partition (dp , dp+1 , . . . , dq ), where di is the degree of vertex i, that is, the number of facets containing it. In the second part of the article, Sections 6–10, we study factorizations of the weighted spanning tree enumerator of Σ under the combinatorial fine weighting xF =

k+1 Y

xi,vi

i=1

(described in more detail in Section 6), where F = {v1 < · · · < vk+1 } is a k-dimensional face of Σ. Thus the term of τk (Σ) corresponding to a particular simplicial spanning tree of Σ contains more precise information than its vertex degrees alone (which can be recovered by further setting xi,j = xj for all i, j). For integer sets A and B as above, we call the ordered pair (A, B) a critical pair of Σ if A ∈ Σ, B 6∈ Σ, and B covers A in the componentwise order. That is, B = {a1 , . . . , ai−1 , ai + 1, ai+1 , . . . , am } for some i ∈ [m]. The long signature of (A, B) is the ordered pair σ ¯ (A, B) = (S, T ), where S = {a1 , . . . , ai−1 } and T = [p, ai ]. The corresponding z-polynomial is defined as z(S, T ) =

1 X XS∪j ↑XS j∈T

where XS = x2S for each S, and the operator ↑ is defined by ↑(xi,j ) = xi+1,j . (See Section 8.1 for more details, and Example 1.7 for an example.) The set of critical pairs is especially significant for a shifted family (and by extension, for a shifted complex). Since a shifted family is just an order ideal with respect to the componentwise partial order , the critical pairs identify the frontier between members and non-members of F in the Hasse diagram of . (See Example 1.7 or [23] for more details.) Thanks to Theorem 1.4, the enumeration of SST’s of a shifted complex reduces to computing the determinant of the reduced combinatorial finely-weighted Laplacian. We show in Section 6 how this computation reduces to the computation of the eigenvalues of the algebraic finely weighted Laplacian. This modification of the combinatorial fine weighting, designed to endow the chain groups of Σ with the structure of an algebraic chain complex, is described in detail in Section 6.3. Its eigenvalues turn out to be precisely the z-polynomials associated with critical pairs. Theorem 1.5. Let Σ be a d-dimensional shifted complex, and let 0 ≤ i ≤ d. Then the eigenvalues of d−i the algebraic finely weighted up-down Laplacian Lud (z(S, T )), where (S, T ) ranges ∆,i are precisely ↑ over all long signatures of critical pairs of i-dimensional faces of Σ. In turn, the z-polynomials are the factors of the weighted simplicial spanning tree enumerator τˆd .

6


Theorem 1.6. Let Σ be a d-dimensional shifted complex with initial vertex p. Then:    Y Y ˜) z(S, T  τˆd (Σ) =  XF˜   X1,p F ∈Λd−1 (S,T )∈¯ σ(∆d )    P Y Y X ˜ S∪j j∈T  XF˜   = XS˜ F ∈Λd−1

(S,T )∈¯ σ(∆d )

where F˜ = F ∪ {p}; ∆ = delp Σ = {F \{p} : F ∈ Σ}; and Λ = linkp Σ = {F : p 6∈ F, F˜ ∈ Σ}. Theorems 1.5 and 1.6 are proved in Sections 8 and 9, respectively. Example 1.7. As an example to which we will return repeatedly, consider the equatorial bipyramid, the two-dimensional shifted complex B with vertices [5] and facets 123, 124, 125, 134, 135, 234, 235. A geometric realization of B is shown in the figure on the left below. The figure on the right illustrates how the facets of B can be regarded as an order ideal. The boldface lines indicate critical pairs. 4 1111 0000 111111 000000 0000 1111 0000 1111 000000 111111 0000 1111 3 0000 1111 000000 111111 000000 111111 0000 1111 000000 111111 0000 1111 000000 111111 000000 111111 0000 1111 000000 111111 0000 1111 000000 000000 111111 0000 1111 2 111111 000000 111111 0000 1111 000000 111111 000000 111111 0000 1111 000000 111111 0000 1 1111 000000 111111 000000 111111 000000 111111 000000 111111

111111 000000 000000 111111 5

146 136

236 145

126

245 235 234

135 125

134 124 123

The Laplacian eigenvalues corresponding to the critical pairs of B are as follows: Critical pair Eigenvalue

(125, 126) (135, 136) (135, 145) (235, 236) (235, 245) z(12, 12345) z(13, 12345) z(1, 123) z(23, 12345) z(2, 123)

To show one of these eigenvalues in more detail, X1,1 X2,1 X3,3 + X1,1 X2,2 X3,3 + X1,1 X2,3 X3,3 + X1,1 X2,3 X3,4 + X1,1 X2,3 X3,5 . z(13, 12345) = X2,1 X3,3 The eigenvalues of this complex are explained in more detail in Section 8.4. Its spanning trees are enumerated in Examples 9.1 (fine weighting) and 9.3 (coarse weighting). We prove Theorem 1.5 by exploiting the recursive structure of shifted complexes. As in [13], we begin by calculating the algebraic finely weighted eigenvalues of a near-cone in terms of the eigenvalues of its link and deletion with respect to its apex (Proposition 7.6). We can then write down a recursive formula (Theorem 8.2) for the nonzero eigenvalues of shifted complexes, thanks to their characterization as iterated near-cones, simultaneously showing that these eigenvalues must be of the form z(S, T ). Finally, we independently establish a recurrence (Corollary 8.8) for the long signatures of critical pairs of a shifted complex, which coincides with the recurrence for the z(S, T ), thus yielding a bijection between nonzero eigenvalues and critical pairs. Corollary 8.10 shows what the eigenvalues look like in coarse weighting. Passing from weighted to unweighted eigenvalues then easily recovers the Duval-Reiner formula for Laplacian eigenvalues of shifted complexes in terms of degree sequences [13, Thm. 1.1]. Similarly, Corollary 9.2 gives the enumeration of SST’s of a shifted complex in the coarse weighting. We are also able to show that the finely-weighted eigenvalues (though not the coarsely-weighted eigenvalues) are enough to recover the shifted complex (Corollary 8.9), or, in other words, that one can “hear the shape” of a shifted complex. Several known results can be obtained as consequences of the general formula of Theorem 1.6.


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• The complete d-skeleton of a simplex is easily seen to be shifted, and applying Theorem 1.6 to such complexes recovers Kalai’s generalization of the Cayley-Pr¨ ufer Theorem. • The one-dimensional shifted complexes are precisely the threshold graphs, an important class of graphs with many equivalent descriptions (see, e.g., [25]). When d = 1, Theorem 1.6 specializes to the weighted spanning tree enumerator for threshold graphs proved by Martin and Reiner [26, Thm. 4] and following from an independent result of Remmel and Williamson [31, Thm. 2.4]. • Thanks to an idea of Richard Ehrenborg, the formula for threshold graphs can be used to recover a theorem of Ehrenborg and van Willigenburg [14], enumerating spanning trees in certain bipartite graphs called Ferrers graphs (which are not in general Laplacian integral). We discuss these corollaries in Section 10. Some classes of complexes that we think deserve further study include matroid complexes, matching complexes, chessboard complexes and color-shifted complexes. The first three kinds of complexes are known to be Laplacian integral, by theorems of Kook, Reiner and Stanton [24], Dong and Wachs [11], and Friedman and Hanlon [17] respectively. Every matroid complex is Cohen-Macaulay [32, §III.3], hence APC, while matching complexes and chessboard complexes are APC for certain values of their defining parameters (see [6]). Color-shifted complexes, which are a common generalization of Ferrers graphs and complete colorful complexes, are not in general Laplacian integral; nevertheless, their weighted simplicial spanning tree enumerators seem to have nice factorizations. It is our pleasure to thank Richard Ehrenborg, Vic Reiner, and Michelle Wachs for many valuable discussions. We also thank Andrew Crites and an anonymous referee for their careful reading of the manuscript. 2. Notation and definitions 2.1. Simplicial complexes. Let V be a finite set. A simplicial complex on V is a family ∆ of subsets of V such that (1) ∅ ∈ ∆; (2) If F ∈ ∆ and G ⊆ F , then G ∈ ∆. The elements of V are called vertices of ∆, and the faces that are maximal under inclusion are called facets. Thus a simplicial complex is determined by its set of facets. The dimension of a face F is dim F = |F | − 1, and the dimension of ∆ is the maximum dimension of a face (or facet). The abbreviation ∆d indicates that dim ∆ = d. We say that ∆ is pure if all facets have the same dimension; in this case, a ridge is a face of codimension 1, that is, dimension dim ∆ − 1. We write ∆i for the set of i-dimensional faces of ∆, and set fi (∆) = |∆i |. The i-skeleton of ∆ is the subcomplex of all faces of dimension ≤ i, [ ∆(i) = ∆j , −1≤j≤i

and the pure i-skeleton of ∆ is the subcomplex generated by the i-dimensional faces, that is, ∆[i] = {F ∈ ∆ : F ⊆ G for some G ∈ ∆i }.

We assume that the reader is familiar with simplicial homology; see, e.g., [18, §2.1]. Let ∆d be a simplicial complex and −1 ≤ i ≤ d. Let R be a ring (if unspecified, assumed to be Z), and let Ci (∆) be the ith simplicial chain group of ∆, i.e., the free R-module with basis {[F ] : F ∈ ∆i }. We denote the simplicial boundary and coboundary maps respectively by ∂∆,i : Ci (∆) → Ci−1 (∆), ∗ ∂∆,i : Ci−1 (∆) → Ci (∆),

where we have identified cochains with chains via the natural inner product. We will abbreviate the subscripts in the notation for boundaries and coboundaries whenever no ambiguity can arise. We will often regard ∂i (resp. ∂i∗ ) as a matrix whose columns and rows (resp. rows and columns) are indexed

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˜ i (∆) = ker(∂i )/ im(∂i+1 ), by ∆i and ∆i−1 respectively. The ith (reduced) homology group of ∆ is H th and the i (reduced) Betti number β˜i (∆) is the rank of the largest free R-module summand of ˜ i (∆). H 2.2. Combinatorial Laplacians. We adopt the notation of [13] for the Laplacian operators (or, equivalently, matrices) of a simplicial complex. We summarize the notation and mention some fundamental identities here. We will often work with multisets (of eigenvalues or of vertices), in which each element occurs with some non-negative integer multiplicity. For brevity, we drop curly braces and commas when working with multisets of integers: for instance, 5553 denotes the multiset in which 5 occurs with multiplicity three and 3 occurs with multiplicity one. The cardinality of a multiset is the sum of the ◦ multiplicities of its elements; thus |5553| = 4. We write a = b to mean that the multisets a and b ◦ ◦ differ only in their respective multiplicities of zero; for instance, 5553 = 55530 = 555300. Of course, ◦ = is an equivalence relation. The union operation ∪ on multisets is understood to add multiplicities: for instance, 5553 ∪ 5332 = 55553332. du tot For −1 ≤ i ≤ dim ∆, define linear operators Lud ∆,i , L∆,i , L∆,i on the vector space Ci (∆) by ∗ Lud ∆,i = ∂i+1 ∂i+1

Ldu ∆,i Ltot ∆,i

= =

∂i∗ ∂i Lud ∆,i

(the up-down Laplacian ), (the down-up Laplacian ),

+ Ldu ∆,i

(the total Laplacian ).

tot ud The spectrum stot i (∆) of L∆,i is the multiset of its eigenvalues (including zero); we define si (∆) and sdu i (∆) similarly. Since each Laplacian operator is represented by a symmetric matrix, it is diagonalizable, so ud du |stot i (∆)| = |si (∆)| = |si (∆)| = fi (∆).

The various Laplacian spectra are related by the identities ◦

du sud i (∆) = si+1 (∆), ◦

ud du stot i (∆) = si (∆) ∪ si (∆)

[13, eqn. (3.6)]. Therefore, each of the three families of multisets {stot i (∆) : − 1 ≤ i ≤ dim ∆},

{sud i (∆) : − 1 ≤ i ≤ dim ∆},

{sdu i (∆) : − 1 ≤ i ≤ dim ∆}

determines the other two, and we will feel free to work with whichever one is most convenient in context. Combinatorial Laplacians and their spectra have been investigated for a number of classes of simplicial complexes. In particular, it is known that chessboard [17], matching [11], matroid [24], and shifted [13] complexes are Laplacian integral, i.e., all their Laplacian eigenvalues are integers. Understanding which complexes are Laplacian integral is an open question. As we will see, Laplacian eigenvalues and spanning tree enumerators are inextricably linked. 3. Simplicial spanning trees In this section, we generalize the notion of a spanning tree to arbitrary dimension using simplicial homology, following Kalai’s idea. Our definition makes sense for any ambient complex that satisfies the relatively mild APC condition. Definition 3.1. Let ∆d be a simplicial complex, and let k ≤ d. A k-dimensional simplicial spanning tree (for short, SST or k-SST) of ∆ is a k-dimensional subcomplex Υ ⊆ ∆ such that Υ(k−1) = ∆(k−1)


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and ˜ k (Υ) = 0, H ˜ k−1 (Υ)| < ∞, |H

(6a)

and ˜ fk (Υ) = fk (∆) − βk (∆) + β˜k−1 (∆).

(6b) (6c)

We write Tk (∆) for the set of all k-SST’s of ∆d , omitting the subscript if k = d. Note that Tk (∆) = Tk (∆(j) ) for all j ≥ k. A zero-dimensional SST is just a vertex of ∆. If ∆ is a 1-dimensional simplicial complex on n vertices—that is, a graph—then the definition of 1-SST coincides with the usual definition of a spanning tree of a graph: namely, a subgraph of ∆ which is connected, acyclic, and has n − 1 edges. Next, we give a few examples in higher dimensions. Example 3.2. If ∆d is a simplicial sphere (for instance, the boundary of a simplicial polytope), then deleting any facet of ∆ while keeping its (d − 1)-skeleton intact produces a d-SST. Therefore |T (∆)| = fd (∆). Example 3.3. In dimension > 1, spanning trees need not be Z-acyclic, merely Q-acyclic. For ˜ 1 (∆, Z) ∼ example, let ∆ be a triangulation of the real projective plane, so that dim ∆ = 2, H = Z/2Z, ˜ and H1 (∆, Q) = 0. Then ∆ satisfies the conditions of Definition 3.1 and is a 2-SST of itself (in fact, the only such). Example 3.4. Consider the equatorial bipyramid B of Example 1.7. A 2-SST of B can be constructed by removing two facets F, F ′ , provided that F ∩ F ′ contains neither of the vertices 4,5. A simple count shows that there are 15 such pairs F, F ′ , so |T2 (B)| = 15. Before proceeding any further, we show that Definition 3.1 satisfies a “two-out-of-three theorem” akin to that for spanning trees of graphs. Proposition 3.5. Let Υ ⊂ ∆d be a k-dimensional subcomplex with Υ(k−1) = ∆(k−1) . Then any two of the conditions (6a), (6b), (6c) together imply the third. Proof. First, note that fℓ (Υ) = fℓ (∆)

for ℓ ≤ k − 1

β˜ℓ (Υ) = β˜ℓ (∆)

and

for ℓ ≤ k − 2.

(7)

Next, we use the standard fact that the Euler characteristic χ(Υ) can be calculated as the alternating sum either of the f -numbers or of the Betti numbers. Thus χ(Υ) =

k X

(−1)i fi (Υ)

i=0

= (−1)k fk (Υ) +

k−1 X

(−1)i fi (∆)

i=0

= (−1)k fk (Υ) + χ(∆) − (−1)k fk (∆)

(8)

and on the other hand, χ(Υ) =

k X

(−1)i β˜i (Υ)

i=0

= (−1)k (β˜k (Υ) − β˜k−1 (Υ)) +

k−2 X (−1)i β˜i (∆) i=0

= (−1)k (β˜k (Υ) − β˜k−1 (Υ)) + χ(∆) − (−1)k (β˜k (∆) − β˜k−1 (∆)).

(9)

10


Equating (8) and (9) gives fk (Υ) − fk (∆) = β˜k (Υ) − β˜k−1 (Υ) − β˜k (∆) + β˜k−1 (∆) or equivalently fk (Υ) − fk (∆) + β˜k (∆) − β˜k−1 (∆) − β˜k (Υ) + β˜k−1 (Υ) = 0.

˜ k (Υ) must be free abelian) and (6b) says that β˜k−1 (Υ) = Since (6a) says that β˜k (Υ) = 0 (note that H 0, the conclusion follows. Definition 3.6. A simplicial complex ∆d is acyclic in positive codimension, or APC for short, if β˜j (∆) = 0 for all j < d. Equivalently, a complex ∆d is APC if it has the homology type of a wedge of zero or more ddimensional spheres. In particular, any Cohen-Macaulay complex is APC. The converse is very far from true, because, for instance, an APC complex need not even be pure. For our purposes, the APC complexes are the “correct” simplicial analogues of connected graphs for the following reason. Proposition 3.7. For any simplicial complex ∆d , the following are equivalent: (1) ∆ is APC. (2) ∆ has a d-dimensional spanning tree. (3) ∆ has a k-dimensional spanning tree for every k ≤ d. Proof. It is trivial that (3) implies (2). To see that (2) implies (1), suppose that ∆ has a d-dimensional ˜ i (∆) = H ˜ i (Υ) = 0 for all i ≤ d − 2. Moreover, spanning tree Υ. Then Υi = ∆i for all i ≤ d − 1, so H in the diagram ∂∆,d

CdS (∆) −−−→ Cd−1 (∆) k Cd (Υ)

∂Υ,d

−−−→

Cd−1 (Υ)

∂∆,d−1

−−−−→ Cd−2 (∆) k ∂Υ,d−1

−−−−→

Cd−2 (Υ)

˜ d−1 (Υ) → we have ker ∂∆,d−1 = ker ∂Υ,d−1 and im ∂∆,d ⊇ im ∂Υ,d , so there is a surjection 0 = H ˜ d−1 (∆), implying that ∆ is APC. H To prove (1) implies (3), it suffices to consider the case k = d, because any skeleton of an APC complex is also APC. We can construct a d-SST Υ by the following algorithm. Let Υ = ∆. If ˜ d (Υ) 6= 0, then there is some nonzero linear combination of facets of Υ that is mapped to zero H by ∂Υ,d . Let F be one of those facets, and let Υ′ = Υ\{F }. Then β˜d (Υ′ ) = β˜d (Υ) − 1 and β˜i (Υ′ ) = β˜i (Υ) for i ≤ d − 2, and by the Euler characteristic formula, we have β˜d−1 (Υ′ ) = β˜d−1 (Υ) ˜ d (Υ) = 0, when Υ is as well. Replacing Υ with Υ′ and repeating, we eventually arrive at the case H a d-SST of ∆. The APC condition is a fairly mild one. For instance, any Q-acyclic complex is clearly APC (and is its own unique SST), as is any Cohen-Macaulay complex (in particular, any shifted complex). 4. Simplicial analogues of the Matrix-Tree Theorem We now explain how to enumerate simplicial spanning trees of a complex using its Laplacian. Throughout this section, let ∆d be an APC simplicial complex on vertex set [n]. For k ≤ d, define X Y ˜ k−1 (Υ)|2 . |H λ, τk = τk (∆) = πk = πk (∆) = 06=λ∈sud k−1 (∆)

Υ∈Tk (∆)

We are interested in the relationships between these two families of invariants. When d = 1, the relationship is given by Theorem 1.1. In the notation just defined, part (1) of that theorem says that τ1 = π1 /n, and part (2) says that τ1 = det Li (i.e., the determinant of the reduced Laplacian obtained from Lud ∆,0 by deleting the row and column corresponding to any vertex i).


11

The results of this section generalize both parts of the Matrix-Tree Theorem from graphs to all APC complexes ∆d . Our arguments are closely based on those used by Kalai [20] and Adin [1] to enumerate SST’s of skeletons of simplices and of complete colorful complexes. We begin by setting up some notation. Abbreviate β˜i = β˜i (∆), fi = fi (∆), and ∂ = ∂∆,d . Let T be a set of facets of ∆ of cardinality fd − β˜d + β˜d−1 = fd − β˜d , and let S be a set of ridges such that |S| = |T |. Define ∆T = T ∪ ∆(d−1) ,

S¯ = ∆(d−1) \ S,

∆S¯ = S¯ ∪ ∆(d−2) ,

and let ∂S,T be the square submatrix of ∂ with rows indexed by S and columns indexed by T . Proposition 4.1. The matrix ∂S,T is nonsingular if and only if ∆T ∈ Td (∆) and ∆S¯ ∈ Td−1 (∆). Proof. We may regard ∂S,T as the top boundary map of the d-dimensional relative complex Γ = ˜ d (Γ) = 0. Consider the long exact sequence (∆T , ∆S¯ ). So ∂S,T is nonsingular if and only if H ˜ d (∆S¯ ) → H ˜ d (∆T ) → H ˜ d (Γ) → H ˜ d−1 (∆S¯ ) → H ˜ d−1 (∆T ) → H ˜ d−1 (Γ) → · · · 0→H

(10)

˜ d (Γ) 6= 0, then H ˜ d (∆T ) and H ˜ d−1 (∆S¯ ) cannot both be zero. This proves the “only if” direction. If H ˜ ˜ ˜ d (∆T ) = 0. Therefore If Hd (Γ) = 0, then Hd (∆S¯ ) = 0 (since dim ∆S¯ = d − 1), so (10) implies H ˜ d−1 (∆T ) is finite. Then (10) ∆T is a d-tree, because it has the correct number of facets. Hence H ˜ d−1 (∆S¯ ) is finite. In fact, it is zero because the top homology group of any complex implies that H must be torsion-free. Meanwhile, ∆S¯ has the correct number of facets to be a (d − 1)-SST of ∆, proving the “if” direction. Proposition 4.2. If ∂S,T is nonsingular, then | det ∂S,T | =

˜ d−1 (∆T )| · |H ˜ d−2 (∆S¯ )| ˜ d−1 (∆T )| · |H ˜ d−2 (∆S¯ )| |H |H = . ˜ ˜ |Hd−2 (∆T )| |Hd−2 (∆)|

Proof. As before, we interpret ∂S,T as the boundary map of the relative complex Γ = (∆T , ∆S¯ ). So ∂S,T is a map from Z|T | to Z|T | , and Z|T | /∂S,T (Z|T | ) is a finite abelian group of order | det ∂S,T |. On the other hand, since Γ has no faces of dimension ≤ d − 2, its lower boundary maps are all zero, so ˜ d−1 (Γ)|. Since H ˜ d−2 (∆T ) is finite, the desired result now follows from the piece | det ∂S,T | = |H ˜ d−2 (∆T ) → 0 ˜ d−1 (∆T ) → H ˜ d−1 (Γ) → H ˜ d−2 (∆S¯ ) → H 0→H of the long exact sequence (10).

(11)

We can now prove the first version of the Simplicial Matrix-Tree Theorem, relating the quantities πd and τd . Abbreviate L = Lud ∆,d−1 . Theorem 1.3 (Simplicial Matrix-Tree Theorem). Let ∆d be an APC simplicial complex. Then: (1) We have πd (∆) =

τd (∆)τd−1 (∆) . ˜ d−2 (∆)|2 |H

(2) Let U be the set of facets of a (d − 1)-SST of ∆, and let LU denote the reduced Laplacian3 obtained by deleting the rows and columns of L corresponding to U . Then τd (∆) =

˜ d−2 (∆)|2 |H det LU . ˜ d−2 (∆U )|2 |H

3A warning: This notation for reduced Laplacians specifies which rows and columns to exclude (in analogy to the notation Li in the statement of Theorem 1.1), in contrast to the notation ∂S,T for restricted boundary maps, which specifies which rows and columns to include.

12


Proof of Theorem 1.3 (1). The Laplacian L is a square matrix with fd−1 rows and columns, and rank fd − β˜d = fd − β˜d + β˜d−1 (because ∆ is APC). Let χ(L; y) = det(yI − L) be its characteristic polynomial (where I is an identity matrix), so that πd (∆), the product of the nonzero eigenvalues ˜ of L, is given (up to sign) by the coefficient of y fd−1 −fd +βd in χ(L; y). Equivalently, X X πd = det LU = det LU (12) S⊂∆d−1 |S|=rank L

S⊂∆d−1 |S|=fd −β˜d

where U = ∆d−1 \S in each summand. By the Binet-Cauchy formula, we have X X ∗ det LU = (det ∂S,T )(det ∂S,T ) = (det ∂S,T )2 . T ⊂∆d |T |=|S|

(13)

T ⊂∆d |T |=|S|

Combining (12) and (13), applying Proposition 4.1, and interchanging the sums, we obtain X X (det ∂S,T )2 πd = T :∆T ∈Td (∆)

S:∆S¯ ∈Td−1 (∆)

and now applying Proposition 4.2 yields X

πd =

T :∆T ∈Td (∆)

 =



X

S:∆S¯ ∈Td−1 (∆)

X

T :∆T ∈Td (∆)

˜ d−1 (∆T )| · |H ˜ d−2 (∆S¯ )| |H ˜ d−2 (∆)| |H 

˜ d−1 (∆T )|2   |H

X

S:∆S¯ ∈Td−1 (∆)

˜ d−2 (∆)|2 |H

!2 

˜ d−2 (∆S¯ )| |H

as desired.

In order to prove the “reduced Laplacian” part of Theorem 1.3, we first check that when we delete the rows of ∂ corresponding to a (d − 1)-SST, the resulting reduced Laplacian has the correct size, namely, that of a d-SST. Lemma 4.3. Let U be the set of facets of a (d − 1)-SST of ∆, and let S = ∆d−1 \U . Then |S| = fd (∆) − β˜d (∆), the number of facets of a d-SST of ∆. Proof. Let Γ = ∆(d−1) . By Proposition 3.5 and the observation (7), |U | = fd−1 (Γ) − β˜d−1 (Γ) + β˜d−2 (Γ) = fd−1 (∆) − β˜d−1 (Γ), so |S| = β˜d−1 (Γ). The Euler characteristics of ∆ and Γ are χ(∆) =

d d X X (−1)i β˜i (∆), (−1)i fi (∆) = i=0

i=0

χ(Γ) =

d−1 X (−1)i β˜i (Γ). (−1)i fi (Γ) =

d−1 X i=0

i=0

By (7), we see that χ(∆) − χ(Γ) = (−1)d fd (∆) = (−1)d β˜d (∆) + (−1)d−1 β˜d−1 (∆) − (−1)d−1 β˜d−1 (Γ) from which we obtain fd (∆) = β˜d (∆) − β˜d−1 (∆) + β˜d−1 (Γ). Since ∆ is APC, we have β˜d−1 (∆) = 0, so |S| = β˜d−1 (Γ) = fd (∆) − β˜d (∆) as desired.


Proof of Theorem 1.3 (2). By the Binet-Cauchy formula, we have X X ∗ (det ∂S,T )(det ∂S,T )= det LU =

13

(det ∂S,T )2 .

T : |T |=|S|

T : |T |=|S|

By Lemma 4.3 and Proposition 4.1, ∂S,T is nonsingular exactly when ∆T ∈ Td (∆). Hence Proposition 4.2 gives !2 X ˜ d−1 (∆T )| · |H ˜ d−2 (∆U )| |H det LU = ˜ d−2 (∆)| |H T :∆ ∈T (∆) T

d

˜ d−2 (∆U )|2 |H = ˜ d−2 (∆)|2 |H

X

˜ d−1 (∆T )|2 = |H

T :∆T ∈Td (∆)

˜ d−2 (∆U )|2 |H τ (∆), ˜ d−2 (∆)|2 d |H

which is equivalent to the desired formula.

˜ d−2 (∆) = 0 (for example, if ∆ is Cohen-Macaulay). Then the two Remark 4.4. Suppose that H versions of Theorem 1.3 assert that det LU πd = , τd = ˜ d−2 (∆U )|2 τd−1 |H from which it is easy to recognize the two different versions of the classical Matrix-Tree Theorem, Theorem 1.1. (A graph is Cohen-Macaulay as a simplicial complex if and only if it is connected.) Moreover, the recurrence τd = πd /τd−1 leads to an expression for τd as an alternating product of eigenvalues: d Y πd πd−2 · · · (−1)d−k τd = = πk . (14) πd−1 πd−3 · · · k=0

This formula is reminiscent of the Reidemeister torsion of a chain complex or CW-complex (although τd is of course not a topological invariant); see, e.g., [35]. Furthermore, (14) is in practice an efficient way to calculate τd . Example 4.5. For the equatorial bipyramid B, we have π0 (B) = 5,

π1 (B) = 5 · 5 · 5 · 3 = 375,

π2 (B) = 5 · 5 · 5 · 3 · 3 = 1125.

These numbers can be checked by computation, and also follow from the Duval-Reiner formula for Laplacian eigenvalues of a shifted complex. Applying the alternating product formula (14) yields 1125 · 5 τ0 (B) = 5, τ1 (B) = 375/5 = 75, τ2 (B) = = 15. 375 Indeed, τ0 (B) is the number of vertices. Cayley’s formula implies that deleting any one edge e from Kn yields a graph with (n − 2)nn−3 spanning trees (because e itself belongs to (n − 1)/ n2 of the spanning trees of Kn ), and the 1-skeleton B(1) is such a graph with n = 5, so τ1 (B) = 75. Finally, we have seen in Example 3.4 that τ2 (B) = 15. 5. Weighted enumeration of simplicial spanning trees We can obtain much finer enumerative information by labeling the facets of a complex with indeterminates, so that the invariant τk becomes a generating function for its SST’s. Let ∆d be an APC simplicial complex, and let ∂ = ∂∆,d . Introduce an indeterminate Q xF for each facet F of maximum dimension, and let XF = x2F . For every T ⊆ ∆d , let xT = F ∈T xF and let XT = x2T . To construct the weighted boundary matrix ∂ˆ from ∂, multiply each column of ∂ˆ by xF , where F is the facet of ∆ corresponding to that column. The weighted coboundary ∂ˆ∗ is the ˆ We can now define weighted versions of Laplacians, the various submatrices of the transpose of ∂. boundary and coboundary matrices used in Section 4, and the invariants πk and τk . We will notate

14


each weighted invariant by placing a hat over the symbol for the corresponding unweighted quantity. ˆ ud Thus π ˆk is the product of the nonzero eigenvalues of L ∆,k−1 , and X ˜ k−1 (Υ)|2 XΥ . |H τˆk = τˆk (∆) = Υ∈Tk (∆)

To recover any unweighted quantity from its weighted analogue, set xF = 1 for all F ∈ ∆d . Proposition 5.1. Let T ⊂ ∆d and S ⊂ ∆d−1 , with |T | = |S| = fd − β˜d . Then det ∂ˆS,T = xT det ∂S,T is nonzero if and only if ∆T ∈ Td (∆) and ∆S¯ ∈ Td−1 (∆). In that case, ˜ d−1 (∆T )| · |H ˜ d−2 (∆S¯ )| ˜ d−1 (∆T )| · |H ˜ d−2 (∆S¯ )| |H |H xT = xT . ± det ∂ˆS,T = ˜ ˜ |Hd−2 (∆T )| |Hd−2 (∆)|

(15)

Proof. The first claim follows from Proposition 4.1, and the second from Proposition 4.2.

It is now straightforward to adapt the proofs of both parts of Theorem 1.3 to the weighted setting. ˆ=L ˆ ud . For convenience, we restate the result. Let L ∆,d−1 Theorem 1.4 (Weighted Simplicial Matrix-Tree Theorem). Let ∆d be an APC simplicial complex. Then: (1) We have π ˆd (∆) =

τˆd (∆)τd−1 (∆) . ˜ d−2 (∆)|2 |H

ˆ U be the reduced Laplacian obtained (2) Let U be the set of facets of a (d− 1)-SST of ∆, and let L ˆ by deleting the rows and columns of L corresponding to U . Then τˆd (∆) =

˜ d−2 (∆)|2 |H Û . det L ˜ d−2 (∆U )|2 |H

Proof. For assertion (1), we use a weighted version of the argument of part (1) of Theorem 1.3. By the Binet-Cauchy formula and Proposition 5.1, we have X X XX ∗ π ˆd = (det ∂ˆT,S )(det ∂ˆS,T ) = (det ∂ˆS,T )2 S⊂∆d−1 T ⊂∆d |T |=|S|

=

S

X

X

X

X

T

(det ∂ˆS,T )2

T :∆T ∈Td (∆) S:∆S¯ ∈Td−1 (∆)

=

T :∆T ∈Td (∆) S:∆S¯ ∈Td−1 (∆)

˜ d−1 (∆T )| · |H ˜ d−2 (∆S¯ )| |H ˜ d−2 (∆)| |H

!2

XT =

τˆd (∆)τd−1 (∆) . ˜ d−2 (∆)|2 |H

The proof of assertion (2) of the theorem is identical to that of part (2) of Theorem 1.3, using Proposition 5.1 instead of Proposition 4.2. Example 5.2. We return to the equatorial bipyramid B of Example 1.7. Weight each facet F = {i, j, k} by the monomial xF = xi xj xk . Let U = {12, 13, 14, 15} be the facets of a 1-SST of B(1) . ˆ U is Then the reduced Laplacian L 0 B B B B @

x2 x3 (x1 + x4 + x5 ) −x2 x3 x4 x2 x3 x4 x2 x3 x5 −x2 x3 x4

−x2 x3 x5 x2 x5 (x1 + x3 ) 0 −x2 x3 x5 0

x2 x3 x4 0 x3 x4 (x1 + x2 ) 0 −x2 x3 x4

x2 x3 x5 −x2 x3 x5 0 x3 x5 (x1 + x2 ) 0

−x2 x3 x4 0 −x2 x3 x4 0 x2 x4 (x1 + x3 )

1 C C C C A


15

and the generating function for 2-SST’s by their degree sequences is X Y deg (i) ˆS = xi B = x31 x32 x33 x24 x25 (x1 + x2 + x3 )(x1 + x2 + x3 + x4 + x5 ) τˆ2 (B) = det L Υ∈T (B) i∈[5]

where degB (i) means the number of facets of B containing vertex i. Setting xi = 1 for every i recovers the unweighted equality τ2 (B) = 15 (see Examples 3.4 and 4.5). 6. Shifted complexes 6.1. General definitions. In the next several sections of the paper, we apply the tools just developed to the important class of shifted complexes. We begin by reviewing some standard facts about shifted complexes and shifted families; for more details, see, e.g., [21]. Let k be an integer. A k-set is a set of integers of cardinality k. A k-family is a set of k-sets (for example, the set of (k − 1)-dimensional faces of a simplicial complex). The componentwise partial order on k-sets of integers is defined as follows: if A = {a1 < a2 < · · · < ak } and B = {b1 < b2 < · · · < bk }, then A B if aj ≤ bj for all j. A k-family F is shifted if B ∈ F and A B together imply A ∈ F. Equivalently, F is shifted if it is an order ideal with respect to the componentwise partial order. A simplicial complex Σ is shifted if Σi is shifted for all i. Accordingly, we may specify a shifted complex by the list of its facets that are maximal with respect to , writing Σ = hF1 , . . . , Fn i. For example, the bipyramid of Example 1.7 is the shifted complex h235i. We will not lose any generality by assuming that the vertex set for every shifted complex we encounter is an integer interval [p, q] = {p, p + 1, . . . , q}; in particular, we will use the symbol p throughout for the vertex with the smallest index. The deletion and link of Σ with respect to p are defined to be the subcomplexes ∆ = delp Σ = {F \{p} : F ∈ Σ}, Λ = linkp Σ = {F : p 6∈ F, F ∪ {p} ∈ Σ}. It is easy to see that the deletion and link of a shifted complex on [p, q] are themselves shifted complexes4 on [p + 1, q]. A complex Σ on vertex set V is called a near-cone with apex p if it has the following property: if F ∈ delp Σ and v ∈ F , then F \{v} ∈ linkp Σ (equivalently, F \{v} ∪ {p} ∈ Σ). It is easy to see that a shifted complex on [p, q] is a near-cone with apex p. Björner and Kalai [5, Theorem 4.3] showed that the Betti numbers of a shifted complex Σ (indeed, of a near-cone) with initial vertex p are given by β˜i (Σ) = |{F ∈ Σi : p 6∈ F, F ∪˙ {p} 6∈ Σ}|. (16) 6.2. The combinatorial fine weighting. Let {xi,j } be a set of indeterminates, indexed by integers i, j. Let k be the field of rational functions in the xi,j with coefficients in C (or in any other field of characteristic zero). Since these indeterminates will often appear squared, we set Xi,j = x2i,j . The combinatorial fine weighting assigns to a multiset of vertices S = {i1 ≤ i2 ≤ · · · ≤ im } the monomials (17) xS = x1,i1 x2,i2 · · · xm,im and XS = X1,i1 X2,i2 · · · Xm,im . Our goal is to describe the generating function X ˜ d−1 (Σ, Z)|2 XΥ |H τˆd (Σ) = Υ∈T (Σ)

of a shifted complex Σ, where, for each simplicial spanning tree Υ, the monomial Y XΥ = XF facets F ∈Υ

4This is also true for the deletion and link with respect to any vertex, not just p, but then the resulting vertex set is no longer a set of consecutive integers. Since we will not have any need to take the deletion and link with respect to any vertex other than p, we won’t worry about that, and instead enjoy the resulting simplicity of specifying the new minimal vertex of the deletion and link.

16


records both the number of facets of Υ containing each vertex of Σ, as well as the order in which the vertices appear in facets. Define the “raising operator” ↑ by ↑xi,j = xi+1,j for i ≤ d and ↑xd+1,j = 0. We extend ↑ linearly and multiplicatively to an operator on all of k. The raising operator can also be applied to a k-linear operator f by the rule (↑f )(V ) =↑(f (↑−1 (V ))) (18) th a for any vector V over k. The a iterate of ↑ is denoted ↑ . The following identities will be useful. Let S˜ = S ∪ {p}, where ∪ denotes the union as multisets, so that the multiplicity of p in S˜ is one more than its multiplicity in S. Then, for all integers a, j, (19a) ↑a x1,p · ↑a+1 xS∪j = ↑a x1,p · ↑xS∪j = ↑a xS∪j ˜ and

↑a xS˜ = xa+1,p = ↑a x1,p . (19b) ↑a+1 xS The same identities hold if x is replaced with X. Now, define the combinatorially finely weighted simplicial boundary map of Σ as the homomorphism ∂ˆ = ∂ˆΣ,i : Ci (Σ) → Ci−1 (Σ) which acts on generators [F ] (for F ∈ Σi ) by X ˆ ]= ∂[F ε(j, F ) ↑d−i xF [F \j]. (20) j∈F

j+1

Here we have set ε(v, F ) = (−1) if v is the j th smallest vertex of F , and ε(v, F ) = 0 if v 6∈ F . ∗ Similarly define the finely weighted simplicial coboundary map ∂ˆ∗ = ∂ˆF,i+1 : Ci (Σ) → Ci+1 (Σ) by X ε(j, F ∪ j) ↑d−i−1 xF ∪j [F ∪ j]. (21) ∂ˆ∗ [F ] = j∈V \F

These maps do not make the chain groups of Σ into an algebraic chain complex, because ∂ˆ∂ˆ and ∂ ∂ do not vanish in general. (We will fix this problem in Section 6.3.) On the other hand, they have combinatorial significance, because we will be able to apply part (2) of Theorem 1.4 to the ˆ ud = ∂ˆd ∂ˆ∗ . This Laplacian may be regarded as a matrix whose finely weighted up-down Laplacian L d rows and columns are indexed by Σd−1 . It is not hard to check that for each F, G ∈ Σd−1 , the ˆ ud is corresponding entry of L  ε(j, H) ε(i, H) XH if H = F ∪ j = G ∪ i ∈ Σ,    P ud XF ∪j if F = G, ˆ )F G = (L (22) j : F ∪j∈Σ    0 otherwise. ˆ∗ ˆ∗

Let U be the simplicial spanning tree of Σ(d−1) consisting of all ridges containing vertex p. (This subcomplex is an SST because it has a complete (d − 2)-skeleton and is a cone over p, hence ˆ ud be the reduced Laplacian obtained from L ˆ ud by deleting the corresponding contractible.) Let L U rows and columns, so that the remaining rows and columns are indexed by the facets of Λ = linkp Σ. ˆ ud . Then part (2) of Theorem 1.4 asserts that τˆd (Σ) = det L U ud ˆ Let N = N (Σ) be the matrix obtained from LU by dividing each row F by ↑xF and dividing each column G by ↑xG . The (F, G) entry of N is thus  XH   if H = F ∪ j = G ∪ i ∈ Σ, ε(j, H) ε(i, H)   ↑x ↑xG  F      P XF ∪j (23) NF G = if F = G,   j : F ∪j∈Σ ↑XF         0 otherwise.


17

Moreover, 

ˆ ud  τˆd (Σ) = det L U =

Y

F ∈Λd−1



↑XF  det N.

(24)

We will shortly see (Lemma 6.3) that N is almost identical to the (full) Laplacian of the deletion ∆ = delp Σ.

6.3. The algebraic fine weighting. The combinatorial fine weighting just defined is awkward to work with directly, because the simplicial boundary and coboundary maps (20) and (21) do not fit together into an algebraic chain complex. Therefore, we introduce a new weighting by Laurent monomials, the algebraic fine weighting, that does give the structure of a chain complex, behaves well with respect to cones and near-cones, and is easy to translate into the combinatorial fine weighting. Definition 6.1. Let ∆d be a simplicial complex on vertices V ⊂ N. The algebraic finely weighted simplicial boundary map of ∆ is the homomorphism ∂ ∆,i : Ci (∆) → Ci−1 (∆) given by

∂ ∆,i [F ] =

X

ε(j, F )

j∈F

↑d−i (xF ) [F \j] ↑d−i+1 (xF \j )

(25)

and similarly the algebraic finely weighted simplicial coboundary map ∂ ∗∆,i+1 : Ci (∆) → Ci+1 (∆) is given by ∂ ∗∆,i+1 [F ] =

X

ε(j, F ∪ j)

j∈V \F

↑d−i−1 (xF ∪j ) [F ∪ j]. ↑d−i (xF )

(26)

We will sometimes drop one or both subscripts when no confusion can arise. By the formula (18), we can apply the raising operator ↑to ∂ and ∂ ∗ by applying it to each matrix entry. That is, for [F ] ∈ Ci (∆) and a ∈ N, we have ↑a ∂ ∆,i [F ] =

X

ε(j, F )

j∈F

↑a ∂ ∗∆,i [F ] =

X

↑d−i+a (xF ) [F \j], ↑d−i+a+1 (xF \j )

ε(j, F ∪ j)

j∈V \F

↑d−i+a−1 (xF ∪j ) [F ∪ j]. ↑d−i+a (xF )

(27) (28)

Lemma 6.2. Let a ∈ N. Then ↑a ∂ ◦ ↑a ∂ = 0 and ↑a ∂ ∗ ◦ ↑a ∂ ∗ = 0. That is, the algebraic finely weighted boundary and coboundary operators induce chain complexes ∂ ∆,i+1

∂ ∆,i

∂∗ ∆,i+1

∂∗ ∆,i

· · · → Ci+1 (∆) −−−−→ Ci (∆) −−−→ Ci−1 (∆) → · · ·

· · · ← Ci+1 (∆) ←−−−− Ci (∆) ←−−− Ci−1 (∆) ← · · ·

18


Proof. Since the matrices that represent the maps ∂ and ∂ ∗ are mutual transposes, it suffices to prove the first assertion. For F ∈ ∆i , we have by (27)   d−i+a X ↑ (x ) F [F \j] ε(j, F ) d−i+a+1 ↑a ∂ ∆,i−1 (↑a ∂ ∆,i ([F ])) = ↑a ∂ ∆,i−1  ↑ (xF \j ) j∈F

=

d−i+a

X

ε(j, F )

↑ (xF ) · ↑a ∂ ∆,i−1 ([F \j]) ↑d−i+a+1 (xF \j )

X

ε(j, F )

X ↑d−i+a+1 (xF \j ) ↑d−i+a (xF ) ε(k, F \j) [F \j\k] ↑d−i+a+1 (xF \j ) ↑d−i+a+2 (xF \j\k )

j∈F

=

j∈F

=

k∈F \j

X X

ε(j, F )ε(k, F \j)

j∈F k∈F \j

=

X

j,k∈F j6=k

↑d−i+a (xF ) [F \j\k] d−i+a+2 ↑ (xF \j\k )

ε(j, F )ε(k, F \j) + ε(k, F )ε(j, F \k)

↑d−i+a (xF ) [F \j\k] ↑d−i+a+2 (xF \j\k )

and it is a standard fact of simplicial homology theory that the parenthesized expression is zero. Define the algebraic finely weighted up-down, down-up, and total Laplacians by ∗ Lud ∆,i = ∂ ∆,i+1 ∂ ∆,i+1 ,

∗ Ldu ∆,i = ∂ ∆,i ∂ ∆,i ,

ud du Ltot ∆,i = L∆,i + L∆,i .

Each of these is a linear endomorphism of Ci (∆), represented by a symmetric matrix, hence diagdu tot ud onalizable. Let sud i (∆), si (∆), and si (∆) denote the spectra (multisets of eigenvalues) of L∆,i , ud du tot tot , sud , sdu , stot when no Ldu ∆,i , and L∆,i respectively. We will use the abbreviations L , L , L confusion can arise. If we regard Lud ∆,d−1 as a matrix with rows and columns indexed by ∆d−1 , then it is not hard to check that its (F, G) entry is  XH   ε(j, H) ε(i, H) if H = F ∪ j = G ∪ i ∈ ∆,   ↑x  F ↑xG      P ud XF ∪j (L∆,d−1 )F G = (29) if F = G,   j : F ∪j∈∆ ↑XF         0 otherwise.

This matrix is almost identical to the matrix N (Σ) defined in (23) when ∆ = delp Σ, as we now explain. Lemma 6.3. Let Σd be a pure shifted complex with initial vertex p, and let Λ = linkp Σ and ∆ = delp Σ. Then   Y Y (X1,p + λ). τˆd (Σ) =  ↑XF  F ∈Λd−1

λ∈sud ∆,d−1

Proof. First, note that L = Lud ∆,d−1 is indexed by the faces of ∆d−1 and N = N (Σ) is indexed by the faces of Λd−1 . These indexing sets coincide because Σ is shifted and pure of dimension d. Second, we show that the off-diagonal entries (the first cases in (23) and (29)) coincide. Suppose that F, G are distinct faces in ∆d−1 = Λd−1 , and that H = F ∪ i = G ∪ j. Suppose that i 6= j and H = F ∪ i = G ∪ j. We must show that H ∈ ∆ if and only if H ∈ Σ. The “only if” direction is


19

immediate because ∆ ⊂ Σ. On the other hand, H = F ∪ G and no element of ∆d−1 contains p. Therefore, if H ∈ Σ, then H ∈ ∆, as desired. Third, we compare the entries on the main diagonals of L and N . Their only difference is that the summand with j = p occurs in the second case of (23), but not in (29). Hence NF F = LF F +

XF ∪p = LF F + X1,p ↑XF

(30)

by (19b). Therefore N = L + X1,p I, where I is an identity matrix of size fd−1 (∆), and det N = χ(−L, X1,p ) =

Y

(X1,p + λ),

λ∈sud ∆,d−1

where χ denotes the characteristic polynomial of −L in the variable X1,p . The lemma now follows from equation (24). The goal of the next two sections is to compute Lud ∆,d−1 .

7. Cones and near-cones A shifted complex is an iterated near-cone, so we want to describe the Laplacian eigenvalues of a near-cone in terms of its base. Before we do so, we must consider the case of a cone. Proposition 7.2 provides the desired recurrence for cones, and Proposition 7.6 for near-cones. 7.1. Boundary and coboundary operators of cones. Let Γ be the simplicial complex with the single vertex 1, and let ∆d be any complex on V = [2, n]. For a face F ∈ ∆, write F˜ = 1 ∪ F . The corresponding cone is Σ = 1 ∗ ∆ = Γ ∗ ∆ = {F, F˜ : F ∈ ∆}. We will make use of the identification Ci (Σ) ∼ = (C−1 (Γ) ⊗ Ci (∆)) ⊕ (C0 (Γ) ⊗ Ci−1 (∆)). By (27) and (28), the (raised) boundary and coboundary maps on Γ are given explicitly by ↑a ∂ Γ [1] = xa+1,1 [∅],

↑a ∂ ∗Γ [1] = 0,

↑a ∂ Γ [∅] = 0,

↑a ∂ ∗Γ [∅] = xa+1,1 [1].

Next, we give explicit formulas for the maps ∂ Σ,i and ∂ ∗Σ,i+1 . How these maps act on a face of Σ depends on whether it is of the form F , for F ∈ ∆i , or F˜ , for F ∈ ∆i−1 . Note that in any case ε(1, F˜ ) = 1, and that for all v ∈ F we have ε(v, F ) = −ε(v, F˜ ). Therefore, ↑d−i+1 (xF ) [∅] ⊗ [F \j] ↑d−i+2 (xF \j ) j∈F   d−i X ↑ (x ) F =↑ ε(j, F ) d−i+1 [∅] ⊗ [F \j] ↑ (xF \j )

∂ Σ,i ([∅] ⊗ [F ]) =

X

ε(j, F )

j∈F

= (id ⊗ ↑∂ ∆,i )([∅] ⊗ [F ]),

(31a)

20


∂ Σ,i ([1] ⊗ [F ]) = ε(1, F˜ )

X ↑d−i+1 (x ˜ ) ↑d−i+1 (xF˜ ) [∅] ⊗ [F ] + ε(j, F˜ ) d−i+2 F [1] ⊗ [F \j] d−i+2 ↑ (xF ) ↑ (xF˜ \j ) j∈F

xd−i+2,1 ↑d−i+2 (xF ) [1] ⊗ [F \j] xd−i+3,1 ↑d−i+3 (xF \j ) j∈F   ↑d−i+1 (xF ) xd−i+2,1  X ↑ [1] ⊗ [F \j] ε(j, F ) d−i+2 = xd−i+2,1 [∅] ⊗ [F ] − xd−i+3,1 ↑ (xF \j ) j∈F xd−i+2,1 d−i+1 = ↑ ∂ Γ ⊗ id − id ⊗ ↑∂ ∆,i−1 ([1] ⊗ [F ]), xd−i+3,1 = xd−i+2,1 [∅] ⊗ [F ] −

X

ε(j, F )

(31b)

X ↑d−i (x ˜ ) ↑d−i (xF ∪j ) ∂ ∗Σ,i+1 ([∅] ⊗ [F ]) = ε(1, F˜ ) d−i+1 F [1] ⊗ [F ] + ε(j, F ∪ j) d−i+1 [∅] ⊗ [F ∪ j] ↑ (xF ) ↑ (xF ) j∈V \F   d−i−1 X ↑ (x ) F ∪j [∅] ⊗ [F ∪ j] ε(j, F ∪ j) = xd−i+1,1 [1] ⊗ [F ]+ ↑ ↑d−i (xF ) j∈V \F d−i ∗ = ↑ ∂ Γ ⊗ id + id ⊗ ↑∂ ∗∆,i+1 ([∅] ⊗ [F ]), (31c) ∂ ∗Σ,i+1 ([1] ⊗ [F ]) =

X

ε(j, F˜ ∪ j)

j∈V \F

=−

X

ε(j, F ∪ j)

j∈V \F

=− =



xd−i+1,1  ·↑ xd−i+2,1

↑d−i (xF˜ ∪j ) ↑d−i+1 (xF˜ ∪j )

[1] ⊗ [F ∪ j]

↑d−i (x1,1 ) ↑d−i+1 (xF ∪j ) [1] d−i+1 ↑ (x1,1 ) ↑d−i+2 (xF )

X

j∈V \F

d−i

ε(j, F ∪ j)

⊗ [F ∪ j] 

↑ (xF ∪j ) [1] ⊗ [F ∪ j] ↑d−i+1 (xF )

xd−i+1,1 ∗ − id ⊗ ↑∂ ∆,i ([1] ⊗ [F ]). xd−i+2,1

(31d)

7.2. Eigenvectors of cones. In order to describe the Laplacian eigenvalues and eigenvectors of 1 ∗ ∆ in terms of those of ∆, we first need some basic facts about the Laplacians of an arbitrary simplicial complex. The following proposition does not depend on fine weighting, and works with any weighted boundary map that satisfies ∂ 2 = 0. ud du du Proposition 7.1. Let Ωd be a simplicial complex, and let −1 ≤ i ≤ d. Let Lud i = LΩ,i , Li = LΩ,i , ∗ ∗ ∂ i = ∂ Ω,i , and ∂ i = ∂ Ω,i . (1) The chain group Ci (Ω) decomposes as a direct sum

Ci (Ω) = Ciud (Ω) ⊕ Cidu (Ω) ⊕ Ci0 (Ω)

(32)

where • Ciud (Ω) has a basis consisting of eigenvectors of Lud i whose eigenvalues are all nonzero, and on which Ldu acts by zero; i • Cidu (Ω) has a basis consisting of eigenvectors of Ldu i whose eigenvalues are all nonzero, and on which Lud acts by zero; and i du 0 • Lud i and Li both act by zero on Ci (Ω). ∗ ud du (2) ker(Li ) = ker(∂ i+1 ) and ker(Li ) = ker(∂ i ). (3) dim Ci0 (Ω) = β˜i (Ω), the ith Betti number of Ω.


21

du tot (4) Each of the spectra sud of Ω has cardinality fi (Ω) as a multiset, and i , si , si ◦

ud sdu i = si−1 , and ◦ stot i =

sud i

∪

◦ sdu i =

(33a) sud i

∪

sud i−1 .

(33b)

du ud Proof. For assertion (1), note that Lud = ∂∂ ∗ ∂ ∗ ∂ = 0 and Ldu = ∂ ∗ ∂∂∂ ∗ = 0. Thus i Li i Li ud du du we may simply take Ci (Ω) and Ci (Ω) to be the spans of the eigenvectors of Lud i and Li with nonzero eigenvalues. For assertion (2), the operators Lud and ∂ ∗i+1 act on the same space, namely Ci (Ω), and they i have the same rank (this is just the linear algebra fact that rank(M M T ) = rank M for any matrix ∗ M ). Therefore, their kernels have the same dimension. Clearly ker(Lud i ) ⊇ ker(∂ i+1 ), so we must have equality. The same argument shows that ker(Ldu i ) = ker(∂ i ). Assertion (3) follows from the calculation

˜ i (Ω, Q) = (dim ker ∂ i ) − (dim im ∂ i+1 ) β˜i (Ω) = dim H = (fi − rank ∂ i ) − (rank ∂ i+1 ) = dim Ci − dim Ciud − dim Cidu = dim Ci0 . For assertion (4), first note that dim Ci (Ω) = fi (Ω), and that the matrix representing each spectrum is symmetric, hence diagonalizable. The identity (33a) is a standard fact in linear algebra, and (33b) is a consequence of the decomposition (32). Proposition 7.2. As in Section 7.1, let ∆d be a simplicial complex on vertex set [2, n], and let Σ = 1 ∗ ∆. Then ◦ ud sud i (Σ) = Xd−i+1,1 + ↑λ : λ ∈ si (∆), λ 6= 0 Xd−i+1,1 ↑µ : µ ∈ sud (∆), µ = 6 0 ∪ Xd−i+1,1 + (34) i−1 Xd−i+2,1 β˜i (∆)

∪ {Xd−i+1,1 }

.

Here the symbol ∪ denotes multiset union, and the superscript in the last line indicates multiplicity. Proof. Throughout the proof, we abbreviate Lud Σ,i by L. All other Laplacians that arise will be specified precisely. ud First, let V ∈ Ciud (∆) be an eigenvector of Lud ∆,i with eigenvalue λ 6= 0. Then ↑L∆,i (↑V ) =↑λ ↑V , and, by Lemma 6.2, 1 (35) ↑∂ ∆,i ↑∂ ∆,i+1 (↑∂ ∗∆,i+1 (↑V )) = 0. ↑∂ ∆,i (↑V ) = ↑λ Using (31a). . . (31d) and (35), we calculate

L ([∅]⊗ ↑V ) = ∂ Σ,i (∂ ∗Σ,i ([∅]⊗ ↑V )) = ∂ Σ,i ↑d−i ∂ ∗Γ [∅]⊗ ↑V + [∅]⊗ ↑∂ ∗∆,i+1 (↑V )

= xd−i+1,1 ∂ Σ,i+1 ([1]⊗ ↑V ) + ∂ Σ,i+1 [∅]⊗ ↑∂ ∗∆,i+1 (↑V ) xd−i+1,1 d−i = xd−i+1,1 ↑ ∂ Γ [1]⊗ ↑V − [1]⊗ ↑∂ ∆,i (↑V ) xd−i+2,1 + [∅]⊗ ↑∂ ∆,i+1 (↑∂ ∗∆,i+1 (↑V )) = (Xd−i+1,1 + ↑λ) ([∅]⊗ ↑V ) .

(36)

Therefore, [∅]⊗ ↑V ∈ Ci (Σ) is an eigenvector of L, with eigenvalue Xd−i+1,1 + ↑λ. Notice that this eigenvalue cannot be zero: since 1 6∈ ∆, no Laplacian eigenvalue of ∆ can possibly equal −Xd−i,1 .

22


Second, let W be an eigenvector in Cidu (∆) with nonzero eigenvalue µ. That is, Ldu ∆i (W ) = = µW , and ∂ ∗∆,i+1 (W ) = 0 by a computation similar to (35). Define

∂ ∗∆,i (∂ ∆,i (W ))

A = [∅]⊗ ↑W,

B = [1]⊗ ↑∂ ∆,i (↑W ).

Note that these are both nonzero elements of Ci (Σ). Then L(A) = ∂ Σ,i+1 ∂ ∗Σ,i+1 ([∅]⊗ ↑W )

= ∂ Σ,i+1 ↑d−i ∂ ∗Γ [∅]⊗ ↑W + [∅]⊗ ↑∂ ∗∆,i+1 (↑W )

= xd−i+1,1 ∂ Σ,i+1 ([1]⊗ ↑W ) xd−i+1,1 d−i [1]⊗ ↑∂ ∆,i (↑W ) = xd−i+1,1 ↑ ∂ Γ [1]⊗ ↑W − xd−i+2,1 Xd−i+1,1 B = Xd−i+1,1 A − xd−i+2,1

(37a)

(37b)

and L(B) = ∂ Σ,i+1 ∂ ∗Σ,i+1 ([1]⊗ ↑∂ ∆,i (↑W )) xd−i+1,1 ∗ = ∂ Σ,i+1 − [1]⊗ ↑∂ ∆,i (↑∂ ∆,i (↑W )) xd−i+2,1 xd−i+1,1 ↑µ · ∂ Σ,i+1 ([1]⊗ ↑W ) =− xd−i+2,1 Xd−i+1,1 Xd−i+1,1 = − ↑µ A + ↑µ B, xd−i+2,1 Xd−i+2,1 where the last step follows by the equality of (37a) and (37b). Letting f = Xd−i+1,1 ,

g=−

Xd−i+1,1 , xd−i+2,1

h=−

↑µ , xd−i+2,1

the calculations above say that L(A) = f A + gB,

L(B) = h(f A + gB),

which implies that L(f A + gB) = f L(A) + gL(B) = f (f A + gB) + gh(f A + gB) = (f + gh)(f A + gB). That is, f A + gB is an eigenvector of L with eigenvalue f + gh = Xd−i+1,1 +

Xd−i+1,1 ↑µ. Xd−i+2,1

As before, this quantity cannot be zero because ∆ does not contain the vertex 1. Third, let Z ∈ Ci0 (∆); we will show that [∅]⊗ ↑Z is an eigenvector of L with eigenvalue Xd−i+1,1 . Indeed, by (2) of Proposition 7.1, we have ∂ ∆,i (Z) = ∂ ∗∆,i+1 (Z) = 0, so that ↑∂ ∆,i (↑Z) =↑∂ ∗∆,i+1 (↑Z) = 0. Therefore L([∅]⊗ ↑Z) = ∂ Σ,i+1 (∂ ∗Σ,i+1 ([∅]⊗ ↑Z)) = ∂ Σ,i+1 (↑d−i ∂ ∗Γ [∅]⊗ ↑Z) + [∅]⊗ ↑∂ ∗∆,i+1 (↑Z)

= xd−i+1,1 ∂ Σ,i+1 ([1]⊗ ↑Z) xd−i+1,1 [1]⊗ ↑∂ ∆,i (↑Z) = xd−i+1,1 · ↑d−i ∂ Γ [1]⊗ ↑Z − xd−i+2 = Xd−i+1,1 ([∅]⊗ ↑Z),

(38)


23

as desired. By assertion (3) of Proposition 7.1, the multiplicity of this eigenvalue is dim Ci0 (∆) = β˜i (∆). ◦ At this point, we have proven that (34) is true if “=” is replaced with “⊇”. On the other hand, we have accounted for dim Ciud (∆) + dim Cidu (∆) + dim Ci0 (∆) = dim Ci (∆) = fi (∆) nonzero eigenvalues in sud i (Σ). Since the preceding calculations hold for all i, we also know fi−1 (∆) ◦ du nonzero eigenvalues of Lud Σ,i−1 (= LΣ,i ). By (33b), we have accounted for all fi (∆) + fi−1 (∆) = tot fi (Σ) = dim Ci (Σ) eigenvalues of LΣ,i . So we have indeed found all the nonzero eigenvalues of L. tot One can obtain explicit formulas for the spectra sdu i (Σ) and si (Σ) by applying (33a) and (33b) to the formula (34); we omit the details.

7.3. Eigenvalues of near-cones. The next step is to establish a recurrence (Proposition 7.6) for the Laplacian eigenvalues of near-cones, in terms of the eigenvalues of the deletion and the link of the apex. Our method is based on that of Lemma 5.3 of [13]. By itself, Proposition 7.6 is not a proper recurrence, in the sense that it computes the eigenvalues of a pure complex in terms of complexes that are not necessarily pure. Therefore, it cannot be applied recursively; the proper recurrence for shifted complexes will have to wait until Theorem 8.2. Since we will be comparing complexes with similar face sets but of different dimensions, we begin by describing how their Laplacian spectra are related. d−i ud Lemma 7.3. Let Σd be a simplicial complex, and let j < i ≤ d. Then sud sj (Σ(i) ). j (Σ) =↑

Proof. The complexes Σ(i) and Σ have the same face sets for every dimension ≤ i, but dim Σ(i) = dim Σ − (d − i). Therefore, the algebraic finely weighted boundary maps and Laplacians of Σ can be obtained from those of Σ(i) by applying ↑d−i , from which the lemma follows. Recall that the pure i-skeleton of Σ is the subcomplex Σ[i] generated by the i-dimensional faces of Σ. ◦

ud Lemma 7.4. Let Σd be a simplicial complex. Then sud d−1 (Σ) = sd−1 (Σ[d] ).

Proof. This result is proved in [12, Lemma 3.2], but we sketch the proof here for completeness. First, observe that Lud d−1 depends only on (d − 1)- and d-dimensional faces. Letting Ω = Σ[d] , we have ud Ωd = Σd and Ωd−1 ⊆ Σd−1 , and indeed Lud Σ,d−1 [F ] = LΩ,d−1 [F ] for any F ∈ Ωd−1 . On the other hand, Σd−1 \Ωd−1 consists precisely of those faces G not contained in any d-dimensional faces of Σ. But Lud Σ,d−1 acts by zero on any such G, and the lemma follows immediately. ◦

d−i ud Corollary 7.5. Let Σd be a simplicial complex. Then sud si−1 (Σ[i] ). i−1 (Σ) = ↑ ◦

d−i ud d−i ud Proof. We have sud si−1 (Σ(i) ) = ↑ d−i sud si−1 (Σ[i] ) by Lemmas 7.3 i−1 (Σ) = ↑ i−1 ((Σ(i) )[i] ) = ↑ and 7.4. (Note that (Σ(i) )[i] = Σ[i] .)

Proposition 7.6. Let Σd be a pure near-cone with apex p, and let ∆ = delp Σ and Λ = linkp Σ be the deletion and link, respectively, of Σ with respect to vertex p. Then ◦ ud sud d−1 (Σ) = X1,p + λ : λ ∈ sd−1 (∆), λ 6= 0 X1,p ∪ X1,p + ↑µ : µ ∈ sud (Λ), µ = 6 0 d−2 X2,p ∪ {X1,p }

β˜d−1 (∆)

.

24


Proof. If Σ is a cone, then the result follows from direct application of Proposition 7.2. (Note that sud d−1 (∆) consists of only 0’s in this case, since ∆ is only (d − 1)-dimensional.) Thus, we may as well assume for the remainder of the proof that Σ is not a cone. In this case, dim ∆ = d and dim Λ = d − 1. It is not difficult to see that ∆(d−1) = Λ, and

(39)

Σ(d) = Σ = (p ∗ ∆)(d) .

(40)

Applying Lemma 7.3 to equation (40) (keeping in mind that dim(p ∗ ∆) = d + 1), and then applying Proposition 7.2, we find that ud ↑sud d−1 (Σ) = sd−1 (p ∗ ∆) ◦ = X2,p + ↑λ : λ ∈ sud d−1 (∆), λ 6= 0 X2,p ud ↑µ : µ ∈ sd−2 (∆), µ 6= 0 ∪ X2,p + X3,p ˜

∪ {X2,p }βd−1 (∆)

so that ud sud d−1 (Σ) = X1,p + λ : λ ∈ sd−1 (∆), λ 6= 0 X1,p µ : µ ∈ sud (∆), µ = 6 0 ∪ X1,p + d−2 X2,p ˜

∪ {X1,p }βd−1 (∆) .

The desired result now follows from applying Lemma 7.3 to equation (39).

8. The Laplacian spectrum of a shifted complex In this section, we explicitly describe the eigenvalues of the algebraic finely weighted Laplacians of a shifted complex Σ. The eigenvalues are Laurent polynomials called z-polynomials, which are in one-to-one correspondence with the critical pairs of the complex: pairs (A, B) such that A ∈ Σ, B 6∈ Σ, and B covers A in the componentwise partial order. The main result, Theorem 1.5, is proved by establishing identical recurrences for the z-polynomials (Theorem 8.2) and critical pairs (Corollary 8.8). 8.1. z -polynomials. Let S and T be multisets of integers. Define a Laurent polynomial z(S, T ) by the formula 1 X XS∪j , (41) z(S, T ) = ↑XS j∈T

where as usual the symbol ∪ denotes multiset union. An example was given at the end of Example 1.7. Proposition 8.1. Let d > i be integers, and let S, T be sets of integers greater than p. Then Xd−i+1,p + ↑d−i z(S, T ) = ↑d−i z(S, T˜)

(42)

and Xd−i+1,p +

Xd−i+1,p d−i+1 ˜ T˜) ↑ z(S, T ) = ↑d−i z(S, Xd−i+2,p

where S˜ = S ∪ {p} and T˜ = T ∪ {p}.

(43)


25

Proof. We will use the identity (19a) repeatedly in the calculations. For (42), observe that Xd−i+1,p + ↑d−i z(S, T ) = Xd−i+1,p +

1

↑d−i+1 XS 

X

↑d−i XS∪j

j∈T



1 Xd−i+1,p · ↑d−i+1 XS + ↑d−i XS∪j  ↑d−i+1 XS j∈T   X 1 ↑d−i X ˜ + ↑d−i XS∪j  = d−i+1 S ↑ XS

=

X

j∈T

=

1 ↑d−i+1 XS d−i

=↑

X

↑d−i XS∪j

j∈T˜

z(S, T˜),

and for (43), observe that Xd−i+1,p +

X Xd−i+1,p Xd−i+1,p d−i+1 ↑d−i+1 XS∪j ↑ z(S, T ) = Xd−i+1,p + Xd−i+2,p Xd−i+2,p · ↑d−i+2 XS j∈T

Xd−i+1,p X d−i+1 = Xd−i+1,p + d−i+1 ↑ XS∪j ↑ XS˜ j∈T   X 1 Xd−i+1,p · ↑d−i+1 X ˜ + = d−i+1 Xd−i+1,p · ↑d−i+1 XS∪j  S ↑ XS˜ j∈T   X 1 ↑d−i X ˜ + = d−i+1 ↑d−i XS∪j ˜  S∪p ↑ XS˜ j∈T

=

1 ↑d−i+1 XS˜

X

↑d−i XS∪j ˜

j∈T˜

˜ T˜). =↑d−i z(S,

Theorem 8.2. Let Σd be a shifted simplicial complex. Then every nonzero eigenvalue of sud i−1 (Σ) has the form of a z-polynomial. Moreover, the spectrum sud (Σ) is determined recursively as follows. i−1 If Σ has no vertices, then sud i−1 (Σ) has no nonzero elements. Otherwise, n o ◦ d−i sud z(S, T˜) : 0 6= z(S, T ) ∈ sud i−1 (Σ) = ↑ i−1 (∆) (44) n o n oβ˜ (∆) d−i ˜∅) i−1 ˜ T˜) : 0 6= z(S, T ) ∈ sud ∪ ↑d−i z(S, (Λ) ∪ ↑ z(∅, i−2 where p is the initial vertex of Σ[i] ; S˜ = S ∪ p; T˜ = T ∪ p; ∆ = delp Σ[i] ; and Λ = linkp Σ[i] .

Proof. The proof is by induction on the number of vertices of Σ. When Σ has no vertices, Σ is either the empty complex with no faces, or the trivial complex whose only face is the empty face. In either case sud i−1 (Σ) has no nonzero elements.

26


We now assume that Σ has at least one vertex. Then ◦ ud sud i−1 (Σ[i] ) = X1,p + λ : λ ∈ si−1 (∆), λ 6= 0 X1,p ∪ X1,p + ↑µ : µ ∈ sud (Λ), µ = 6 0 i−2 X2,p β˜i−1 (∆)

∪ {X1,p + 0}

= X1,p + z(S, T ) : z(S, T ) ∈ sud i−1 (∆), z(S, T ) 6= 0 X1,p z(S, T ) : z(S, T ) ∈ sud (Λ), z(S, T ) = 6 0 ∪ X1,p + i−2 X2,p β˜i−1 (∆)

∪ {X1,p + z(∅, ∅)}

◦

n o = z(S, T˜) : 0 6= z(S, T ) ∈ sud i−1 (∆) n o ˜ T˜) : 0 6= z(S, T ) ∈ sud (Λ) ∪ z(S, i−2 n oβ˜i−1 (∆) . ∪ z(∅, ˜ ∅)

The =-equivalence above is by Proposition 7.6. The following equality is justified by the identity z(∅, ∅) = 0 and induction on the number of vertices, since ∆ and Λ each have one fewer vertex than Σ. Note that λ and µ are each replaced by z(S, T ), with no raising operator, because dim ∆ ≤ i and dim Λ = i − 1. The final equality comes from Proposition 8.1. The result now follows from Corollary 7.5. 8.2. Critical pairs. Throughout this section, let F be a k-family of sets of integers, and let p be the smallest integer occurring in any element of F . Definition 8.3. A critical pair for F is an ordered pair (A, B), where A = {a1 < a2 < · · · < ak } and B = {b1 < b2 < · · · < bk } are sets of integers such that A ∈ F, B 6∈ F, and B covers A in componentwise order. That is, bi = ai + 1 for exactly one i, and bj = aj for all j 6= i. (Note that bi need not be in the vertex set of F .) The signature of (A, B) is the set of vertices σ(A, B) = {a1 , . . . , ai−1 , ai }. The long signature is the ordered pair of sets σ ¯ (A, B) = (S, T ), where S = {a1 , . . . , ai−1 } and T = {j : p ≤ j ≤ ai }. The multisets of signatures and long signatures of critical pairs of F are denoted σ(F ) and σ ¯ (F ) respectively. As described in the Introduction and Example 1.7, critical pairs are especially significant for shifted simplicial complexes. We will soon see that the critical pairs of a shifted complex are in bijection with the eigenvalues of its algebraic finely weighted Laplacian. Definition 8.4. The degree of vertex v in the family F is degF (v) = |{F ∈ F : v ∈ F }|. Proposition 8.5. Let F be a shifted family. Then degF (v) − degF (v + 1) counts the number of signatures of F whose greatest element is v. Proof. Let S = {F ∈ F : v ∈ F } and T = {F ∈ F : v + 1 ∈ F }. Partition S = S1 ∪˙ S2 ∪˙ S3 and T = T1 ∪˙ T2 ∪˙ T3 as follows: S1 = {F ∈ F : v, v + 1 ∈ F }, S2 = {F ∈ F : v ∈ F, v + 1 6∈ F, F \{v} ∪˙ {v + 1} ∈ F }, S3 = {F ∈ F : v ∈ F, v + 1 6∈ F, F \{v} ∪˙ {v + 1} 6∈ F }, T1 = {F ∈ F : v, v + 1 ∈ F }, T2 = {F ∈ F : v 6∈ F, v + 1 ∈ F, F \{v + 1} ∪˙ {v} ∈ F }, T3 = {F ∈ F : v 6∈ F, v + 1 ∈ F, F \{v + 1} ∪˙ {v} 6∈ F }.


27

Then S1 = T1 , and there is an obvious bijection between S2 and T2 . Since F is shifted, T3 = ∅, so degF (v) − degF (v + 1) = |S| − |T | = |S3 |. Finally, if F ∈ S3 , then (F, G) is a critical pair, where G = F \{v} ∪˙ {v + 1}. For each such critical pair, the greatest element of the signature is v. Conversely, if (A, B) is a critical pair whose signature’s greatest element is v, then A ∈ S3 . Corollary 8.6. Let Σd be a pure shifted simplicial complex with initial vertex p. Then degΣd (p) − degΣd (p + 1) = β˜d−1 (delp Σ). Proof. Just as in the proof of Proposition 8.5 above, degΣd (p) − degΣd (p + 1) = |S3 |, where S3 = {F ∈ Σd : p ∈ F, p + 1 6∈ F, F \{p} ∪˙ {p + 1} 6∈ Σd }. There is a bijection between S3 and the set S3′ = {G ∈ (delp Σ)d−1 : p + 1 6∈ G, G ∪˙ {p + 1} 6∈ delp Σ} given by G = F \{p}. Then, by equation (16), β˜d−1 (delp Σ) = |S3′ |. Combining the four displayed equations yields the desired result.

Proposition 8.7. Let Σ be a shifted complex with initial vertex p. Let ∆ = delp Σ and Λ = linkp Σ. Then σ(Σi ) = σ(∆i ) ∪ {p ∪˙ F : F ∈ σ(Λi−1 )} ∪ {p}degΣi (p)−degΣi (p+1) , where ∪ denotes multiset union. Proof. The multiplicity of the signature {p} follows from Proposition 8.5. Suppose that A ∈ ∆i , B 6∈ ∆i , and B covers A in the componentwise partial order. Then A ∈ Σi and p 6∈ A. Since p 6∈ A and A ≺ B, we conclude that p 6∈ B, and so B 6∈ Σi . Hence we have a one-to-one map of multisets σ(∆i ) → σ(Σi ). On the other hand, suppose that A ∈ Λi−1 , B 6∈ Λi−1 , and B covers A in the componentwise partial order. Then p 6∈ A and A˜ = A ∪˙ {p} ∈ Σi . The critical pair (A, B) of Λi−1 gives rise to the ˜ B) ˜ of Σi , and it is clear that σ(A, ˜ B) ˜ = σ(A, B) ∪˙ {p}. Furthermore, B ˜ 6∈ Σi because critical pair (A, B 6∈ Λi−1 . Hence we have a one-to-one map of multisets {p ∪˙ F : F ∈ σ(Λi−1 )} → σ(Σi ). Now we must show, conversely, that every signature F 6= {p} of Σi arises in one of these two ways. Let F be such a signature of Σi , with critical pair (A, B). So A ∈ Σi ; B 6∈ Σi (so B 6∈ ∆i ); and B covers A in the componentwise partial order. First, if p 6∈ F , then p 6∈ A and so A ∈ ∆i . Thus (A, B) is a critical pair for ∆i . Second, if p ∈ F , then F = F ′ ∪˙ {p} for some F ′ 6= ∅. In this case, we have p ∈ A and p ∈ B for the critical pair (A, B) whose signature is F . Indeed, p ∈ F directly implies that p ∈ A. If p 6∈ B, then F = σ(A, B) = {p}. Accordingly, let A′ = A\{p} and B ′ = B\{p}. Then A′ ∈ Λi−1 and B 6∈ Σi , so B ′ 6∈ Λi−1 , and (A, B) is a critical pair for Λi−1 . ¯ (Σi ) is Corollary 8.8. Let Σd be a shifted simplicial complex, and let i ≤ d. Then the multiset σ determined by the following recurrence. If Σ has no vertices, then σ ¯ (Σi ) = ∅.

28


Otherwise, σ ¯ (Σi ) = {(S, T˜) : (S, T ) ∈ σ ¯ (∆i )} ˜ T˜) : (S, T ) ∈ σ ∪ {(S, ¯ (Λi−1 )} ∪ {(∅, ˜∅)}

(45)

β˜i−1 (∆)

where p is the initial vertex of Σ[i] ; S˜ = S ∪ p; T˜ = T ∪ p; ∆ = delp Σ[i] ; and Λ = linkp Σ[i] . Proof. If σ(A, B) = {p}, then σ ¯ (A, B) = (∅, {p}) = (∅, ˜∅). So the third term in (45) arises from applying Corollary 8.6 to the pure i-dimensional shifted complex Σ[i] . (Indeed, Σ[i] is shifted when Σ is, or even just when Σi is.) For the first two terms in (45), the only hard part is to note that if the first vertex of Σi is p, then p + 1 is the first vertex of ∆i and Λi−1 (unless ∆i = ∅, but in that case we don’t have to worry about the first set). This accounts for the T˜’s. Even though Proposition 8.7 above defines ∆ and Λ somewhat differently than here, it is not a problem because (delp Σ)i = (delp Σ[i] )i and (linkp Σ)i−1 = (linkp Σ[i] )i−1 . Then, since ∆ and Λ only appear as ∆i and Λi−1 , it doesn’t matter whether we set them to be the deletion and link of Σ or of Σ[i] . 8.3. Theorem 1.5 and its consequences. We can finally prove Theorem 1.5, which characterizes the Laplacian spectra sud i−1 (Σ) in terms of z-polynomials of critical pairs. In the notation we have developed, the theorem can be restated as follows: Theorem 1.5. Let Σd be a shifted simplicial complex. Then, for each 0 ≤ i ≤ d, ◦

d−i sud {z(S, T ) : (S, T ) ∈ σ ¯ (Σi )}. i−1 (Σ) = ↑

Proof. Simply note that the recursions in Theorem 8.2 and Corollary 8.8 are identical.

(46)

One corollary to Theorem 1.5 is that you can “hear the shape” of a shifted complex (Corollary 8.9), but only if your ears are fine enough (Remark 8.11). d Corollary 8.9. A shifted complex Σd is completely determined by its spectra {sud i−1 (Σ)}i=0 . d Proof. By Theorem 1.5, the spectra {sud i−1 (Σ)}i=0 determine the z-polynomials z(S, T ), and it is easy to see that the long signature (S, T ) can be recovered from z(S, T ). Furthermore the (short) signature is even more easily recovered from the long signature. When F = {v1 < . . . < vi } is a face of Σ that is -maximal, then (F, F \{vi } ∪˙ {vi+1 }) is a critical pair, with (short) signature F . Thus, among all the (short) signatures of Σ, we will find all -maximal faces. Furthermore, every (short) signature is a face of Σ, by definition. Thus, the union ∪F ∈σ(Σ) {G : G F } of all -order ideals will yield all the non-empty faces of Σ.

In fact, if Σd is a pure shifted complex, then it is determined uniquely by its top Laplacian spectrum sud d−1 (Σ). Specializing (or “coarsening”) the algebraic fine weighting, we obtain as another corollary to Theorem 1.5 Duval and Reiner’s description of the unweighted Laplacian eigenvalues of a shifted complex [13, Thm. 1.1], as we now explain. The coarse weighting is obtained from the algebraic fine weighting by omitting all first subscripts, i.e., replacing xi,j = xj and Xi,j = Xj . (Thus the monomial corresponding to a facet or set of facets records the degree of each vertex, but forgets the information about the order of vertices in facets.) Note that if T = [1, t], then every z-polynomial z(S, T ) specializes to the linear form Et = X1 + · · · + Xt in the coarse weighting. For a partition λ (a weakly decreasing list of positive integers), let Eλ be the multiset in which each part i of λ is replaced by Ei . Recall that the conjugate of λ is the partition λ′ in which each part t occurs with multiplicity λt − λt+1 .


29

Corollary 8.10. Let Σd be a shifted simplicial complex on vertices [n]. Then ◦

sûd d−1 (Σ) = E(degΣ

d

)′

where the left-hand side denotes the multiset of coarsely weighted Laplacian eigenvalues, and the right-hand side is the conjugate of the partition (degΣd (1), . . . , degΣd (n)). ◦

Proof. Theorem 1.5 and the preceding discussion imply that sûd d−1 (Σ) is =-equivalent to the multiset in which Et occurs with multiplicity equal to the number of critical pairs (A, B) of Σd such that t = max σ(A, B). By Proposition 8.5, that multiplicity is degΣd (t) − degΣd (t + 1). The result now follows from the definition of conjugate partition.

Passing to the unweighted setting by setting xi = 1 for all i recovers the theorem of Duval and Reiner [13, Thm. 1.1], which states that the Laplacian eigenvalues of a shifted complex Σ are given by the conjugate of the partition degΣd . Remark 8.11. Duval and Reiner also showed [13, Example 10.2] that there are two non-isomorphic 2-dimensional shifted complexes with the same degree sequence. Corollary 8.10 then shows that, in contrast to Corollary 8.9, the coarsely-weighted eigenvalues are not enough to determine a shifted complex. 8.4. An example: the equatorial bipyramid. To illustrate Theorems 8.2 and 1.5, we calculate the top-dimensional up-down Laplacian spectrum of the bipyramid B = h235i (see Examples 1.7, 3.4 and 5.2). In our recursive calculation, we will encounter the subcomplexes B2 , . . . , B7 of B = B1 shown in the following figure, and B8 , the simplicial complex whose only face is the empty face. 4 1111 0000 111111 000000 0000 1111 0000 1111 000000 111111 0000 1111 3 0000 1111 000000 111111 0000 1111 000000 111111 0000 1111 000000 111111 000000 111111 0000 1111 000000 111111 0000 1111 000000 111111 000000 111111 0000 1111 000000 111111 2 0000 1111 000000 111111 000000 111111 0000 1111 000000 111111 0000 1111 000000 111111 000000 111111 000000 111111 000000 1 111111 111111 000000 000000 111111 000000 111111 5

4 111111 000000 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 2 3 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 5

B1

B2

4 4

4

3

3

4 2

3

5 5

5

5

5

B3

B4

B5

B6

B7

Observe that • • • • • • •

B1 B2 B3 B4 B5 B6 B7

is a near-cone = 2 ∗ B4 ; is a near-cone = 3 ∗ B6 ; is a near-cone is a near-cone = 5 ∗ B8 .

with apex 1, del1 B1 = B2 , and link1 B1 = B3 ; with apex 2, del2 B3 = B4 , and link2 B3 = B5 ; with apex 3, del3 B5 = B6 , and link3 B5 = B8 ; with apex 4, del4 B6 = B7 , and link4 B6 = B8 ; and

The critical pairs, signatures, and nonzero top-dimensional Laplacian eigenvalues of the bipyramid B = B1 and its subcomplexes B2 , . . . , B7 are listed in the following table. This information can be obtained either recursively (using Theorem 8.2 repeatedly) or bijectively (from Theorem 1.5).

30


Subcomplex B7 B6 B5 B4

Dimension 0 0 0 1

B3

1

B2

2

B1

2

Vertices {5} {4, 5} {3, 4, 5} {3, 4, 5}

Critical pairs Signatures Eigenvalues (5, 6) 5 z(∅, 5) (5, 6) 5 z(∅, 45) (5, 6) 5 z(∅, 345) (35, 45) 3 z(∅, 3) (35, 36) 35 z(3, 345) {2, 3, 4, 5} (25, 26) 25 z(2, 2345) (35, 36) 35 z(3, 2345) (35, 45) 3 z(∅, 23) {2, 3, 4, 5} (235, 236) 235 z(23, 2345) (235, 245) 23 z(2, 23) {1, 2, 3, 4, 5} (125, 126) 125 z(12, 12345) (135, 136) 135 z(13, 12345) (135, 145) 13 z(1, 123) (235, 236) 235 z(23, 12345) (235, 245) 23 z(2, 123) ◦

We also see from the above table that the coarsely-weighted eigenvalues sûd 1 (B1 ) are =-equivalent to E5 , E5 , E5 , E3 , E3 (each E5 coming from a z(S, 12345) in B1 , and each E3 coming from a z(S, 123) in B1 ), corresponding to the transpose of the degreee sequence of the facets, 55533. (In this case, both the degree sequence and its transpose are 55533.) 9. Enumerating spanning trees of shifted complexes We now translate Theorem 1.5 from the algebraic to the combinatorial fine weighting, in order to obtain a factorization of the finely weighted spanning tree enumerator τˆ(Σ) of a shifted complex Σ. Recall that the long signature σ ¯ (F ) of a family F is the multiset of long signatures of its critical pairs. Theorem 1.6. Let Σd be a shifted complex with initial vertex p. Then:    Y Y ˜ z(S, T )  τˆd (Σ) =  XF˜   X1,p F ∈Λd−1 (S,T )∈¯ σ(∆d )    P Y Y X ˜ S∪j j∈T  = XF˜   XS˜ F ∈Λd−1

(47)

(48)

(S,T )∈¯ σ(∆d )

where F˜ = F ∪ p, ∆ = delp Σ, and Λ = linkp Σ.

Proof. Since the complex Σ is APC, its spanning trees are precisely those of its pure d-skeleton Σ[d] . Similarly, passing from Σ to Σ[d] does not affect ∆d or Λd−1 . Therefore, we may assume without loss of generality that Σ is pure. By Lemma 6.3 and Theorem 1.5, we have    Y Y m τˆd (Σ) =  (X1,p + z(S, T )) X1,p ↑XF   F ∈linkp Σ

(S,T )∈¯ σ(∆d )

ud where m is the number of zero eigenvalues of Lud ¯ (∆d ) ∆,d−1 . Since L∆,d−1 acts on ∆d−1 , but has σ nonzero eigenvalues (including multiplicity), m = |∆d−1 | − |¯ σ (∆d )| = |Λd−1 | − |¯ σ (∆d−1 )|, since


∆d−1 = Λd−1 by equation (39). Thus,    Y |Λd−1 |  −|¯ σ (∆ )| τˆd (Σ) = X1,p (↑XF ) X1,p d  F ∈Λd−1

=

Y

(X1,p ↑XF )

F ∈Λd−1

Y

(S,T )∈¯ σ(∆d )

Y

(S,T )∈¯ σ(∆d )

X1,p + z(S, T ) . X1,p

31



X1,p + z(S, T )

Equation (47) now follows from equations (19a) and (42). Equation (48) then follows from the definition of z-polynomial (41) and from (19a) again, because P P z(S, T˜) 1 j∈T˜ XS∪j j∈T˜ XS∪j = = . X1,p X1,p ↑XS XS˜ Example 9.1. We return to our running example, the equatorial bipyramid B. Here d = 2 and p = 1, and in the notation of Section 8.4, we have ∆ = B2 and Λ = B3 . Moreover, σ ¯ (∆d ) = {(2, 23), (23, 2345)}. Hence equation (48) yields X12 + X22 + X23 X123 + X223 + X233 + X234 + X235 τˆ(B) = X123 X124 X134 X125 X135 . X12 X123 Note that this is a genuine polynomial (not just a Laurent polynomial) in the indeterminates Xi,j . ˜ = 1 ∗ ∆, Corollary 9.2. Let Σd be a shifted complex with initial vertex 1. Let ∆ = del1 Σ, ∆ ˜ Λ = link1 Σ, and Λ = 1 ∗ Λ. Then, in the coarse weighting, Y ′ ˜ ˜ (Ei /X1 )(deg ∆d+1 )i , τˆd (Σ) = X deg Λd i

where Ei = X1 + · · · + Xi and, for a partition λ, we set X λ :=

Q

i

Xiλi . ˜

Proof. Upon coarsening the weighting, the first product in (48) becomes X deg Λd and the second product becomes P Y Y j∈T˜ Xj E|T |+1 /X1 . = X1 (S,T )∈¯ σ(∆d )

(S,T )∈¯ σ(∆d )

We now claim that Y

E|T |+1 /X1 =

(S,T )∈¯ σ(∆d )

Y

E|T | /X1 .

˜ d+1 ) (S,T )∈¯ σ (∆

Indeed, by Proposition 8.7, ˜ d+1 ) = {1 ∪˙ F : F ∈ σ(∆d )} ∪ {1}m σ(∆ ˜ and dim(del1 (∆)) ˜ = dim ∆ < d + 1. It now follows that for some m, since ∆ = link1 (∆) Y Y (E|T |+1 /X1 )(X1 /X1 )m , E|T | /X1 = ˜ d+1 ) (S,T )∈¯ σ (∆

(S,T )∈¯ σ(∆d )

implying the claim. Finally, by Proposition 8.5 and the definition of conjugate partition, we obtain Y Y Y ′ ˜ (t+1) deg (t)−deg∆ ˜ d+1 (Ei /X1 )(deg ∆d+1 )i . E|T | /X1 = = (Et /X1 ) ∆˜ d+1 ˜ d+1 ) (S,T )∈¯ σ (∆

t

i

32


˜ d = {123, 124, 125, 134, 135}. Example 9.3. Once again, let B be the equatorial bipyramid. Here Λ Its degree sequence is 53322. Thus the monomial factor in Corollary 9.2 is X123 X124 X125 X134 X135 = X15 X23 X33 X42 X52 . ˜ d+1 = {1234, 1235}. Its degree sequence is 22211, with conjugate 53. The product Meanwhile, ∆ factor in Corollary 9.2 is therefore (E5 /X1 )(E3 /X1 ) = (X1 + X2 + X3 + X4 + X5 )(X1 + X2 + X3 )/X12 . Putting these terms together yields τˆd (Σ) = X13 X23 X33 X42 X52 (X1 + X2 + X3 + X4 + X5 )(X1 + X2 + X3 ) which matches Example 5.2. 10. Corollaries We conclude by showing how several known tree enumerators—for skeletons of simplices, threshold graphs, and Ferrers graphs—can be recovered from our results. 10.1. Skeletons of simplices. Let Σ be the d-skeleton of the simplex on vertices [n], so that the [n] set of facets of Σ is d+1 , the set of all subsets of [n] of cardinality d + 1. Note that Σ is generated as a shifted complex by the single facet [n − d, n]. The critical pairs of Σ are [n − 1] (A ∪ {n}, A ∪ {n + 1}) : A ∈ d and the corresponding long signatures are [n − 1] σ ¯ (Σd ) = (A, [n]) : A ∈ . d Setting Λ = link1 Σ and ∆ = del1 Σ, we have [2, n] Λd−1 = ; d [2, n] ∆d = ; d+1 [2, n − 1] σ ¯ (∆d ) = (B, [2, n]) : B ∈ . d Applying equation (48), we obtain 



 Y   τˆd (Σ) =  X ˜ C   C⊆[2,n] |C|=d

Y

Pn



XB∪j  .  X˜

j=1

B⊆[2,n−1] |B|=d

B

˜ The denominators in the second product cancel the factors XC˜ in the first product with n 6∈ C, ˜ leaving only those for which n ∈ C. Therefore,             n  Y  Y    Y  Y X     z(B, [n]) X = X τˆd (Σ) =  X   ˜ C B∪j C   .    C⊆[2,n]  B⊆[2,n−1] j=1  B⊆[2,n−1]  C⊆[n] n∈C |C|=d

|B|=d

1,n∈C |C|=d+1

|B|=d


33

Passing to the coarse weighting by setting Xi,j = Xj for every i, j, we obtain n−3 n−2 n−2 n−3 τˆd (Σ) = (X1 Xn )( d−1 ) (X2 · · · Xn−1 )( d−2 ) (X2 · · · Xn−1 )( d−1 ) (X1 + · · · + Xn )( d ) n−2 n−2 = (X1 · · · Xn )( d−1 ) (X1 + · · · + Xn )( d ) ,

which is Theorem 3′ of Kalai’s paper [20]. Furthermore, setting Xi = 1 for all i recovers Kalai’s n−2 generalization of Cayley’s formula: τd (Σ) = n( d ) . 10.2. Threshold graphs. A threshold graph is a one-dimensional shifted complex Σ. For simplicity, we assume that the vertex set of Σ is [1, n]. We may also assume that Σ is connected, so that every vertex is adjacent to vertex 1. Martin and Reiner [27, Theorem 4] found a factorization of the combinatorially finely weighted spanning tree enumerator of Σ, which may be stated in our notation5 as: Σ)′v n−1 Y (deg X X{v,j} . (49) τˆ(Σ) = X{1,n} v=2

j=1

A somewhat more general result was obtained independently by Remmel and Williamson [31, Theorem 2.4]. We will show how this formula can be recovered from Theorem 1.6. The first product in equation (48) is just X{1,2} X{1,3} · · · X{1,n} . For the second product, we must identify the critical pairs of ∆ = del1 Σ. Note that ∆ is a threshold graph with vertices [2, n]. As is often the case with threshold graphs (and their degree sequences, which we will soon encounter), we need to sort vertices by their relation to the size of the Durfee square of Σ, the largest square that fits in the Ferrers diagram of its degree sequence. The side length m of the Durfee square is the largest number such that {m, m + 1} is an edge of Σ. If m = 1, then Σ is a star graph, the Ferrers diagram of its degree sequence is a hook, and equations (48) and (49) both easily reduce to τˆd (Σ) =

n Y

X{1,v} .

v=2

Therefore, we henceforth assume that m ≥ 2. Note that every edge has at least one endpoint ≤ m (because {m+1, m+2} 6∈ Σ, and that edge is the unique -minimal edge with both endpoints > m). For each vertex v of Σ, let w(v) = max{u : {u, v} ∈ Σ}. Note that if v ≤ m, then {v, m + 1} {m, m + 1} ∈ Σ, so w(v) ≥ m. On the other hand, if v > m, then w(v) < v. Lemma 10.1. The critical pairs of ∆ = del1 Σ are as follows. For each v1 ∈ [2, m], there is a “type I” critical pair ({v1 , v2 }, {v1 , v2 + 1}) where v2 = w(v1 ). For each v2 ∈ [m + 2, w(2)], there is a “type II” critical pair ({v1 , v2 }, {v1 + 1, v2 }), where v1 = w(v2 ). Furthermore, every critical pair is of one of these forms. Proof. It is immediate from the definition of w(v) that each such pair is critical. Suppose now that (A, B) is a critical pair of ∆, with A = {v1 < v2 }. Then either B = {v1 , v2 + 1} or B = {v1 + 1, v2 }. If B = {v1 , v2 + 1}, then we have already observed that A has at least one endpoint in [2, m]. In particular, v1 ≤ m, and the pair (A, B) is of type I. If B = {v1 + 1, v2 }, then A ∈ ∆ and B 6∈ ∆, so by definition v1 = w(v2 ). Moreover, m + 2 ≤ v2 (because v1 + 1 ≤ v2 − 1, so {v2 − 1, v2 } B 6∈ ∆) and v2 ≤ w(2) (because {2, v2 } A, so {2, v2 } ∈ ∆). Hence the pair (A, B) is of type II. 5Note the distinction between the variable X , which corresponds to vertex j as the ith smallest vertex in a face, i,j

and the quadratic monomial X{i,j} , which corresponds to the edge {i, j}, and which equals X1,i X2,j if i ≤ j but equals X1,j X2,i if j ≤ i.

34


If (A, B) is a critical pair of type I, then σ(A, B) = {v1 , v2 } and σ ¯ (A, B) = ({v1 }, [2, v2 ]). If (A, B) is a critical pair of type II, then σ(A, B) = {v1 } and σ ¯ (A, B) = (∅, [2, v1 ]). Therefore, formula (48) yields w(2) Pw(v2 ) n m Pw(v1 ) Y Y Y j=1 X{j} j=1 X{v1 ,j} τˆd (Σ) = X{1,v} X X{1} {1,v1 } v2 =m+2 v=2 v1 =2 Pw(v2 ) n m Pw(v1 ) n Y Y Y j=1 X{v1 ,j} j=1 X{j} = X{1,v} X X{1} {1,v1 } v=2 v1 =2 v2 =m+2 ! ! Pw(v1 ) Pw(v2 ) m n Y Y j=1 X{v1 ,j} j=1 X{j} = X{1,n} . (50) X{1,v1 } X{1,v2 −1} X{1,v1 } X{1} v =2 v =m+2 1

2

The second equality follows because w(v) = 1 whenever v > w(2), and the third equality comes from redistributing most of the first product among the other two. Now, when v2 > m + 1, we have Pw(v2 ) w(v2 ) w(v2 ) w(v2 ) X X X1,1 X2,v2 −1 X j=1 X{j} X{1,v2 −1} X1,j = = X1,j X2,v2 −1 = X{j,v2 −1} X{1} X1,1 j=1 j=1 j=1 since j ≤ w(v2 ) ≤ m < v2 − 1. Thus we may rewrite (50) as    w(v2 ) n m w(v X X1 ) Y Y X{j,v2 −1}  X{1,n} . X{v1 ,j}   τˆd (Σ) = 

(51)

v2 =m+2 j=1

v1 =2 j=1

If v1 ≤ m, then w(v1 ) > v1 , so w(v1 ) has degree at least v1 , as does every vertex less than w(v1 ). On the other hand, {v1 , w(v1 ) + 1} 6∈ Σ, so vertex w(v1 ) + 1 has degree less than v1 , as does every vertex greater than w(v1 ) + 1. Therefore, (deg Σ)′v1 = w(v1 ). Similarly, if v2 > m + 1, then ({w(v2 ), v2 )}, {w(v2 ) + 1, v2 }) is a critical pair, so w(v2 ) has degree at least v2 − 1, as does every vertex less than w(v2 ). On the other hand, vertex w(v2 ) + 1 has degree less than v2 − 1, as does every vertex greater than w(v2 ) + 1. Therefore, (deg Σ)′v2 −1 = w(v2 ). Using these observations to rewrite (51) in terms of the partition (deg Σ)′ recovers the MartinReiner formula (49).

10.3. From threshold graphs to Ferrers graphs. Let λ = (λ1 ≥ · · · ≥ λℓ ) be a partition. The corresponding Ferrers graph is the bipartite graph with vertices x1 , . . . , xλ1 , y1 , . . . , yℓ and edges {xi yj : i ≤ λj }. That is, the vertices correspond to rows and columns of the Ferrers diagram of λ, and the edges to squares appearing in the diagram. Ehrenborg and van Willigenburg [14] considered Ferrers graphs and (among other results) described how a certain weighted spanning tree enumerator splits into linear factors. Another proof of their formula can be obtained from the foregoing formulas for threshold graphs, as we now explain. The key idea is due to Richard Ehrenborg. Let G be a connected threshold graph on vertices [n], and let m be the side length of the Durfee square of G. Then the vertices 1, 2, . . . , m are pairwise adjacent, while m + 1, . . . , n are pairwise nonadjacent. Moreover, if m + 1 ≤ i < j ≤ n, then every neighbor of j is a neighbor of i. Construct a graph F by deleting all edges ij such that i, j ≤ m. Then F is a Ferrers graph; furthermore, all Ferrers graphs can be constructed in this way. Thus, if we begin with the weighted enumerator for G and set to zero all indeterminates corresponding to edges between vertices 1, 2, . . . , m, we recover the weighted enumerator for the corresponding Ferrers graph F . Specifically, Theorem 1.6 yields  T  n−1 i (G) Y dX  τˆ(G) = X1,1 Xn,2 Xmin(i,r),1 Xmax(i,r),2  (52) i=2

r=1


35

(this is also [27, Theorem 4]). Breaking up the product in (52) around the parameter m gives    dT m m i (G) X Y X τ (G) = X1,1 Xn,2   Xmin(i,r),1 Xmax(i,r),2  × Xmin(i,r),1 Xmax(i,r),2 + i=2

r=m+1

r=1

 

n−1 Y

dT i (G)

X

i=m+1 r=1

dTi (G)



Xmin(i,r),1 Xmax(i,r),2  .

This expression is well defined because ≥ m whenever i ≤ m. If i ≤ m and r ≤ m, then max(i, r) ≤ m, and these are exactly the terms we wish to set to zero. Therefore,    T T m dX n−1 i (G) i (G) Y Y dX τ (F ) = X1,1 Xn,2  Xi,1 Xr,2   Xmin(i,r),1 Xmax(i,r),2  . (53) i=2 r=m+1

i=m+1 r=1

For i ≥ m + 1, we have dTi (G) < m < i. Thus r < i for r ≤ dTi (G), and (53) yields    T T n−1 m dX n−1 m i (G) i (G) Y dX Y Y Y Xi,2 Xr,1  Xr,2   τ (F ) = X1,1 Xn,2  Xi,1 i=2

i=2 r=m+1

i=m+1 r=1

i=m+1



= (X1,1 X2,1 . . . Xm,1 )(Xm+1,2 Xm+2,2 . . . Xn,2 ) 

T

m dX i (G) Y

i=2 r=m+1



Xr,2  

n−1 Y

dT i (G)

X

i=m+1 r=1

By construction, the vertex degrees in G and F are related by the formula ( degG (i) − m if 1 ≤ i ≤ m, degF (i) = degG (i) if m + 1 ≤ i ≤ n.



Xr,1  . (54)

To simplify the notation, set Xr,1 = xr and yr−m = Xr,2 = yr−m . (From this perspective, the two partite sets of F correspond to the indeterminates {x1 , . . . xm } and {y1 , . . . yn−m }. Therefore,    T T (F2 ) (F1 ) m diX n−m Y Y diX τ (F ) = (x1 . . . xm )(y1 . . . yn−m )  yr   xr  i=2

r=1

i=2

r=1

which is Theorem 2.1 of [14].

References [1] Ron Adin, Counting colorful multi-dimensional trees, Combinatorica 12, no. 3 (1992), 247–260. [2] Eric Babson and Isabella Novik, Face numbers and nongeneric initial ideals, Electron. J. Combin. 11, no. 2 (2004/06), Research Paper #R25, 23 pp. (electronic) [3] Matthias Beck and Serkan Ho¸sten, Cyclotomic polytopes and growth series of cyclotomic lattices, Math. Res. Lett. 13, no. 4 (2006), 607–622. [4] L.W. Beineke and R.E. Pippert, Properties and characterizations of k-trees, Mathematika 18 (1971), 141–151. [5] Anders Bj¨ orner and Gil Kalai, An extended Euler-Poincar´ e theorem, Acta Math. 161, no. 3–4 (1988), 279–303. ˇ [6] A. Bj¨ orner, L. Lov´ asz, S.T. Vre´ cica, and R.T. Zivaljevi´ c, Chessboard complexes and matching complexes, J. London Math. Soc. (2) 49 (1994), no. 1, 25–39. [7] Ethan Bolker, Simplicial geometry and transportation polytopes, Trans. Amer. Math. Soc. 217 (1976), 121–142. [8] B´ ela Bollob´ as, Modern Graph Theory. Graduate Texts in Mathematics, 184. Springer, New York, 1998. [9] Arthur Cayley, A theorem on trees, Quart. J. Math. 23 (1889), 376–378. [10] A.K. Dewdney, Higher-dimensional tree structures, J. Comb. Theory Ser. B 17 (1974), 160–169. [11] Xun Dong and Michelle L. Wachs, Combinatorial Laplacian of the matching complex, Electron. J. Combin. 9 (2002), no. 1, Research Paper #R17, 11 pp. (electronic) [12] Art M. Duval, A relative Laplacian spectral recursion, Electron. J. Combin. 11 (2004/06), no. 2, Research Paper #R26, 19 pp. (electronic) [13] Art M. Duval and Victor Reiner, Shifted simplicial complexes are Laplacian integral, Trans. Amer. Math. Soc. 354, no. 11 (2002), 4313–4344.

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[14] Richard Ehrenborg and Stephanie van Willigenburg, Enumerative properties of Ferrers graphs, Discrete Comput. Geom. 32 (2004), no. 4, 481–492. [15] Sara Faridi, The facet ideal of a simplicial complex, Manuscripta Math. 109, no. 2 (2002), 159–174. ¨ ˇ [16] Miroslav Fiedler and Jiˇr´ı Sedl´ aˇ cek, Uber Wurzelbasen von gerichteten Graphen, Casopis Pˇ est. Mat. 83 (1958), 214–225. [17] Joel Friedman and Phil Hanlon, On the Betti numbers of chessboard complexes, J. Algebraic Combin. 8 (1998), 193–203. [18] Allen Hatcher, Algebraic Topology. Cambridge University Press, Cambridge, 2002. Also available online at http://www.math.cornell.edu/∼hatcher/AT/ATpage.html. [19] J¨ urgen Herzog and Enzo Maria Li Marzi, Bounds for the Betti numbers of shellable simplicial complexes and polytopes, Commutative algebra and algebraic geometry (Ferrara), 157–167, Lecture Notes in Pure and Appl. Math., 206, Dekker, New York, 1999. [20] Gil Kalai, Enumeration of Q-acyclic simplicial complexes, Israel J. Math. 45, no. 4 (1983), 337–351. [21] Gil Kalai, Algebraic shifting, Computational commutative algebra and combinatorics (Osaka, 1999), 121–163, Adv. Stud. Pure Math., 33, Math. Soc. Japan, Tokyo, 2002. ¨ [22] G. Kirchhoff, Uber die Aufl¨ osung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Str¨ ome gef¨ uhrt wird, Ann. Phys. Chem. 72 (1847), 497–508. [23] Caroline Klivans, Obstructions to shiftedness, Discrete Comput. Geom. 33 (2005), 535–545. [24] W. Kook, V. Reiner, and D. Stanton, Combinatorial Laplacians of matroid complexes, J. Amer. Math. Soc. 13 (2002), 129–148. [25] N.V.R. Mahadev and U.N. Peled, Threshold graphs and related topics. Annals of Discrete Mathematics, 56. North-Holland Publishing Co., Amsterdam, 1995. [26] Jeremy L. Martin and Victor Reiner, Cyclotomic and simplicial matroids, Israel J. Math. 150 (2005), 229–240. [27] Jeremy L. Martin and Victor Reiner, Factorization of some weighted spanning tree enumerators, J. Combin. Theory Ser. A 104, no. 2 (2003), 287–300. [28] Gregor Masbaum and Arkady Vaintrob, A new matrix-tree theorem, Int. Math. Res. Not. 2002, no. 27, 1397– 1426. [29] J.W. Moon, Counting Labeled Trees. Canadian Mathematical Monographs, No. 1, Canadian Mathematical Congress, Montreal, 1970. [30] Gerald Reisner, Cohen-Macaulay quotients of polynomial rings, Adv. Math. 21 (1976), no. 1, 30–49. [31] Jeffrey B. Remmel and S. Gill Williamson, Spanning trees and function classes, Electron. J. Combin. 9, no. 1 (2002), Research Paper #R34, 24 pp. (electronic) [32] Richard P. Stanley, Combinatorics and Commutative Algebra, 2nd edn., Birkh¨ auser, Boston, 1996. [33] Richard P. Stanley, Enumerative Combinatorics, volume I. Cambridge Studies in Advanced Mathematics, 49. Cambridge University Press, 1997. [34] Richard P. Stanley, Enumerative Combinatorics, volume II. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, 1999. [35] Vladimir Turaev, Introduction to combinatorial torsions, Lectures in Mathematics ETH Z¨ urich, Birkh¨ auser Verlag, Basel, 2001. Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968-0514 Departments of Mathematics and Computer Science, The University of Chicago, Chicago, IL 60637 Department of Mathematics, University of Kansas, Lawrence, KS 66047

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